Recent zbMATH articles in MSC 30https://zbmath.org/atom/cc/302023-02-24T16:48:17.026759ZWerkzeugBook review of: F. Bracci et al., Continuous semigroups of holomorphic self-maps of the unit dischttps://zbmath.org/1502.000742023-02-24T16:48:17.026759Z"Shoikhet, David"https://zbmath.org/authors/?q=ai:shoikhet.davidReview of [Zbl 1441.30001].Systolic length of triangular modular curveshttps://zbmath.org/1502.110692023-02-24T16:48:17.026759Z"Schein, Michael M."https://zbmath.org/authors/?q=ai:schein.michael-m"Shoan, Amir"https://zbmath.org/authors/?q=ai:shoan.amirThe authors of this article present a method for obtaining explicit upper bounds for the systolic length of certain Riemann surfaces using computations with quaternion algebras over over totally real fields. The Riemann surfaces considered are obtained as quotients by certain subgroups of hyperbolic triangle groups. Interestingly, the latter Fuchsian groups are, in general, \textit{not} arithmetic groups. The authors' method relies on the fact that the uniformising Fuchsian group of a given Riemann surface of the type that they consider embeds into a very specific subgroup (modulo the centre) of an explicit quaternion algebra, and on the fact that the systolic length is related the trace of some hyperbolic element in the uniformising group. Careful tabulations of upper bounds, computed according to the authors' procedure, are presented.
Reviewer: Gautam Bharali (Bangalore)Joint approximation by non-linear shifts of Dirichlet \(L\)-functionshttps://zbmath.org/1502.110912023-02-24T16:48:17.026759Z"Laurinčikas, Antanas"https://zbmath.org/authors/?q=ai:laurincikas.antanas"Šiaučiūnas, Darius"https://zbmath.org/authors/?q=ai:siauciunas.dariusSummary: In the paper, a theorem on the simultaneous approximation of a collection of analytic functions by non-linear shifts of Dirichlet \(L\)-functions \((L(s + i t_\tau^{\alpha_1}, \chi_1), \ldots, L(s + i t_\tau^{\alpha_r}, \chi_r))\) is obtained. Here \(t_\tau\) is the Gram function, \(\alpha_1, \ldots, \alpha_r\) are fixed different positive numbers, and \(\chi_1, \ldots, \chi_r\) are arbitrary Dirichlet characters. Also, an example of approximation by a certain composition of the above shifts is given.Strata of differentials of the second kind, positivity and irreducibility of certain Hurwitz spaceshttps://zbmath.org/1502.140212023-02-24T16:48:17.026759Z"Mullane, Scott"https://zbmath.org/authors/?q=ai:mullane.scottÉtant donné \(g\geq0\) et une partition \(\mu=(m_{1},\dots,m_{n})\) de \(2g-2\) avec \(m_{i}\in\mathbb{Z}\), on définit la strate \(\mathcal{H}(\mu)\) paramétrant les paires \((X;\omega)\) où \(X\) est une surface de Riemann de genre \(g\) et \(\omega\) est une différentielle abélienne telle que \(\operatorname{div}(\omega)=\sum m_{i}p_{i}\). Lorsqu'au moins un des \(m_{i}\) est négatif on peut considérer le sous espace \(\mathcal{H}_{Z}(\mu)\) constitué des différentielles de seconde espèce, i.e. dont tous les résidus sont nuls. Les espaces \(\mathcal{H}_{Z}(\mu)\) sont une sorte d'intermédiaire entre les strates de différentielles abéliennes et les espaces d'Hurwitz. L'article propose deux applications de ces lieux, une dans chaque direction.
La première application est de montrer que la projection de la fermeture de \(\mathcal{H}_{Z}(\mu)\) à \(\bar{\mathcal{M}}_{0,n'}\) est \(F\)-nef. De plus l'auteur calcule la classe de ces projections pour tous les genres dans le cas où ils sont diviseurs et que la projection n'oublie que des zéros simple.
La seconde application est de montrer l'irréductibilité des espaces d'Hurwitz des revêtements de \(\mathbb{P}^{1}\) de degré \(d\), de genre \(g\) dont toutes les ramifications sont pures (la préimage contient un unique point) à l'exception d'au plus un point de ramification.
Un point clé de la preuve de la première application est de déterminer la fermeture de \(\mathcal{H}_{Z}(\mu)\) dans la compactification de la variété d'incidence des strates décrite dans [\textit{M. Bainbridge} et al., Duke Math. J. 167, No. 12, 2347--2416 (2018; Zbl 1403.14058)]. Cela permet d'étendre la méthode introduite par l'A. dans [\textit{S. Mullane}, Mich. Math. J. 67, No. 4, 839--889 (2018; Zbl 1412.14020)] au calcule des classes de diviseurs dans le cas considéré. La seconde application est obtenue en identifiant les espaces d'Hurwitz avec certains lieux de différentielles exactes de genre zéro.
L'article est bien écrit et contient les rappels nécessaires pour sa compréhension. Toutefois une certaine familiarité avec les strates de différentielles et la géométrie birationnelle de l'espace des modules des courbes est souhaitable pour profiter pleinement de ce très intéressant article.
Reviewer: Quentin Gendron (Ciudad de México)Relative stability conditions on Fukaya categories of surfaceshttps://zbmath.org/1502.140442023-02-24T16:48:17.026759Z"Takeda, Alex"https://zbmath.org/authors/?q=ai:takeda.alex-aSince Mumford introduced the concept of slope-stability of holomorphic bundles on a Riemann surface, when he built geometric invariant theory to construct moduli spaces, there has been many expansions and generalizations. Bridgeland defined a stability condition on triangulated categories, inspired by Douglas' \(\Pi\)-stability condition on BPS D-branes on a Calabi-Yau manifold in string theory considering slope-stability condition under variations of Kähler class, [\textit{T. Bridgeland}, Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)]. A Bridgeland-stability condition \(\sigma=(Z,\mathcal{P})\) on a triangulated category \(\mathcal{D}\) is a pair of a group homomorphism \(Z:K(\mathcal{D})\to \mathbb{C}\) from the Grothendieck group of the category to the group of complex numbers and an \(\mathbb{R}\)-graded collection \(\mathcal{P}:=\cup_{\phi\in \mathbb{R}} \mathcal{P}(\phi)\) of objects of \(\mathcal{D}\), which consists of full additive subcategories of \(\mathcal{D}\), satisfying the 4 axioms, see Definition 1.1 of [\textit{T. Bridgeland}, Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)]. One of interesting properties of the space of Bridgeland-stability conditions on a triangulated category is that it carries a natural topology and indeed it is a complex manifold. So one could get an invariant of a triangulated category or of the underlying geometry, for example, Joyce's Hall algebra [\textit{D. Joyce}, Adv. Math. 215, No. 1, 153--219 (2007; Zbl 1134.14007)], Kontsevich-Soibelmans's motivic Hall algebra [\textit{M. Kontsevich} and \textit{Y. Soibelman}, ``Stability structures, motivic Donaldson-Thomas invariants and cluster transformations'', Preprint, \url{arXiv:0811.2435}], with extension of theories.
The space of stability conditions on the partially wrapped Fukaya category of a graded marked Riemann surface are known to be related to the moduli space of quadratic differentials [\textit{T. Bridgeland} and \textit{I. Smith}, Publ. Math., Inst. Hautes Étud. Sci. 121, 155--278 (2015; Zbl 1328.14025)] (considering quiver structures), or to the moduli space of flat structures on the given Riemann surface [\textit{F. Haiden} et al., Publ. Math., Inst. Hautes Étud. Sci. 126, 247--318 (2017; Zbl 1390.32010)] under homeomorphisms. A flat surface \((\Sigma,\Omega)\) is a pair of a Riemann surface \(\Sigma\) and a holomorphic 1-form \(\Omega\) on it. Denote \(\widehat{x}\), the set of zeroes of the holomorpic 1-form \(\Omega\). Then the holomorphic 1-form \(\Omega\) gives rise to a flat Riemannian metric on the complement \(\Sigma\setminus \widehat{x}\) of the zero-sets in the Riemann surface, with cone singularities on \(\widehat{x}\), with straight line flow \(\eta\) and vise versa. The straight line flow \(\eta\), or horizontal foliation provides a grading on \(\Sigma\) and the cone singularities give markings on the boundaries of the induced Riemann surface with boundary. Partially wrapped Fukaya category of a graded marked Riemann surface consists of Lagrangian submanifolds, with conical ends on the marked boundaries up to isotopy, as objects and Lagrangian Floer complexes as morphisms. Every (oriented) arcs, embedded(immersed) interval with ends on marked boundaries or embedded(immersed) circles, on a Riemann surface is a (graded) Lagrangian submanifold of the Riemann surface, considering a volume form as a symplectic structure. Using Dehn twists and Polterovich surgery under partially wrapping, we can drive objects in a complex of nice-shaped indecomposable objects, see Section 3.2 and Corollary 24.
Integration \(H_{1}(\Sigma,\partial \Sigma;\mathbb{Z}_{\eta})\otimes_{\mathbb{Z}}H^{1}_{dR}(\Sigma,\partial \Sigma;\mathbb{C}_{\eta})\to \mathbb{C}\) gives rise to periods of flat surfaces, where \(\mathbb{Z}_{\eta}\) is \(\mathbb{Z}\) tensor with a locally constant sheaf from the canonical double cover of \((\Sigma,\eta)\), \(\mathbb{C}_{\eta}:=\mathbb{Z}_{\eta}\otimes \mathbb{C}\). The period map \(Z:K_{0}(\mathcal{F}(\Sigma))\to H_{1}(\Sigma,\partial \Sigma;\mathbb{Z}_{\eta})\to \mathbb{C}\) satisfies the axiom of Bridgeland-stability conditions, [\textit{F. Haiden} et al., Publ. Math., Inst. Hautes Étud. Sci. 126, 247--318 (2017; Zbl 1390.32010)].
In the paper under review, the author constructed stability conditions on partially wrapped Fukaya categories of graded marked Riemann surfaces in a functorial manner, in other words, in a way of cutting-and-gluing a Riemann surface with flat structure. The author defined the set of relative stability conditions, where stability conditions are defined over extended surface: Let \(S\) be a Riemann surface with an embedded interval \(\gamma\) which connects two adjacent marked boundary intervals \(M\), \(M'\) and runs parallel to the unmarked boundary interval between them. A relative stability condition on the pair \((S, \gamma)\) consists of:
\begin{itemize}
\item an integer \(n\geq 2\), and
\item a stability condition \(\widetilde{\sigma}\in \text{Stab}(\mathcal{F}(S\cup_{\gamma}\Delta_{n}))\), where \(S\cup_{\gamma}\Delta_{n})\) is the extended surface obtained by gluing a disk \(\Delta_{n}\), with \(n\)-marked boundary intervals, to \(S\) at \(\gamma\) along one of its unmarked boundaries.
\end{itemize}
Then the author carefully chose the reduced arc system to get an equivalence relation on the set of relative stability conditions. In the section 4.4, there is an interesting example of other equivalences, from the non-reduced arc system, where the quotient space fails to be Hausdorff. The main result of the paper is that the quotient space is Hausdorff and can be glued to the space of global stability conditions: When we decompose a marked surface \(\Sigma\) into two surfaces \(\Sigma=\Sigma_{L}\cup_{\gamma} \Sigma_{R}\) glued along arcs \(\gamma\). There is a subset \(\Gamma\subset \text{RelStab}(\Sigma_{L},\gamma)\times \text{RelStab}(\Sigma_{R},\gamma)\) and continuous maps \[\text{Stab}(\mathcal{F}(\Sigma))\xrightarrow{\text{cut}_{\gamma}}\Gamma\xrightarrow{\text{glue}_{\gamma}} \text{Stab}(\mathcal{F}(\Sigma))\] that are inverse homeomorphisms, when restricted to the locus of stability conditions whose stable objects are all supported on intervals.
At the end of the paper, the author casts a question how one could extend the definition of relative stability conditions to wrapped Fukaya categories of higher-dimensional symplectic manifolds: What is the basic building block of stability conditions that can be glued to the global stability conditions? On one hand, Weinstein manifolds, a class of non-compact symplectic manifolds which closely relate to Stein manifolds, admit Morse-theoretic decomposition by Weinstein handle attachments. It is also known that the wrapped Fukaya category of a Weinstein manifold can be computed by gluing cosheaves of categories of its skeletons of Weinstein handles by Ganatra-Pardon-Shende and there is combinatorial models of corresponding constructible sheaves by Nadler. On the other hand, considering that stability conditions of Fukaya category of a Riemann surface are based on understanding on Teichmüller dynamics and on the complete classification of admissible Lagrangians, one would need higher dimensional analogues of them.
Reviewer: Dahye Cho (Stony Brook)Platonic solids and high genus covers of lattice surfaceshttps://zbmath.org/1502.140722023-02-24T16:48:17.026759Z"Athreya, Jayadev S."https://zbmath.org/authors/?q=ai:athreya.jayadev-s"Aulicino, David"https://zbmath.org/authors/?q=ai:aulicino.david"Hooper, W. Patrick"https://zbmath.org/authors/?q=ai:hooper.w-patrick"Randecker, Anja"https://zbmath.org/authors/?q=ai:randecker.anjaLes solides de Platon, le tétraèdre, le cube, l'octaèdre, le dodécaèdre et l'icosaèdre, sont parmi les objets mathématiques les plus étudiés. Toutefois, l'étude des géodésiques sur ces solides est restée jusqu'à présent relativement sommaire. Le résultat principal de cet article est de montrer qu'il existe 31 classes d'équivalences de géodésiques fermées sur le dodécaèdre. De plus, les auteurs reprouvent qu'il n'existe pas de géodésiques fermées sur les autres solides.
La preuve de ces résultats est un petit bijoux dont les étapes principales sont les suivantes. Les solides de Platon possèdent une structure métrique plate d'holonomie non-triviale. Cette métrique est induite par la structure d'une \(k\)-différentielle \(\xi\) sur la sphère de Riemann. On peut donc considérer la surface de translation \(S\) obtenue en développant cette métrique: c'est un revêtement ramifié de degré \(k\) de la sphère de Riemann qui correspond au revêtement canonique associé à \(\xi\). Les géodésiques sur un solide de Platon correspondent aux géodésiques sur son revêtement. Depuis les travaux fondateurs de Veech un énorme travail a été entrepris pour comprendre ces géodésiques. En particulier, les revêtements des solides de Platon sont des revêtements réguliers des \(p\)-gones et doubles \(p\)-gones de Veech. Les auteurs s'appuient donc sur le travail [\textit{W. A. Veech}, Geom. Funct. Anal. 2, No. 3, 341--379 (1992; Zbl 0760.58036)] et le package FlatSurf de Sage pour étudier les courbes de Teichmüller associées aux revêtements. Le résultat principal est alors obtenu grâce à la relation entre la géométrie des courbes de Teichmüller et les géodésiques sur les surfaces de translation (cette relation est expliquée en autre dans [\textit{M. Möller}, in: Proceedings of the international congress of mathematicians 2018, ICM 2018, Rio de Janeiro, Brazil, August 1--9, 2018. Volume III. Invited lectures. 2017--2034 (2018; Zbl 1447.32021)]).
Cet article se termine par quatre appendices. Deux sont particulièrement intéressants. Le premier au sujet du fonctionnement du package FlatSurf dans Sage. Le second est une vue historique par Anja Randecker sur les travaux du début du 20ème siècle de Paul Stäckel et Carl Rodenberg.
Reviewer: Quentin Gendron (Ciudad de México)Schottky spaces and universal Mumford curves over \(\mathbb{Z}\)https://zbmath.org/1502.140732023-02-24T16:48:17.026759Z"Poineau, Jérôme"https://zbmath.org/authors/?q=ai:poineau.jerome"Turchetti, Daniele"https://zbmath.org/authors/?q=ai:turchetti.danieleA Schottky group is a free, finitely generated subgroup of \(\mathrm{PGL}_2(\mathbb{C})\). These groups uniformize the Riemann surfaces in the sense that every connected complex analytic curve is the quotient of an open dense subset \(\mathcal{O} \subset \mathbb{C}\) by the action of a Schottky group which has \(\mathcal{O}\) as the region of discontinuity. In the past '70s, D. Mumford started the study of non-archimedean theory of Schottky uniformization. He defined the concept of non-archimedean Schottky group and showed that these groups act on some open subset of the projective line, and the quotient identifies to a projective curve. Also, he characterized the curves that admit Schottky uniformization. V. Berkovich gave a further step in the '80s, defining non-archimedean analytic spaces as spaces of absolute values. If \(k\) is a non-archimedean field, the group \(\mathrm{PGL}_2(k)\) acts on the Berkovich projective line \(\mathbb{P}_k^{1,\mathrm{an}}\), and the Schottky uniformization of Mumford curves holds as in the complex case.
In the article under review, the authors deal with both archimedean and non-archimedean theories of Schottky uniformization. In order to get a common way of study for both theories, they define a moduli space \(\mathcal{S}_g\) of Schottky groups of rank \(g\) as a Berkovich space over \(\mathbb{Z}\). That definition generalizes the Schottky spaces both over \(\mathbb{C}\) and over non-archimedean fields. The space \(\mathcal{S}_g\) is proved (Theorem 4.3.4) to be open in \(\mathbb{A}_{\mathbb{Z}}^{3g-3,\text{an}}\), an affine Berkovich space over \(\mathbb{Z}\). Then, the authors study the connection between \(\mathcal{S}_g\) and the moduli space of curves. If \(\text{Out}(F_g)\) is the group of outer automorphisms of the free group with \(g\) generators, and \(\mathcal{S}_g^{\mathrm{na}}\) is the set of non-archimedean points of \(\mathcal{S}_g\), the action of \(\text{Out}(F_g)\) on \(\mathcal{S}_g^{\mathrm{na}}\) is proper and has finite stabilizers (Corollary 5.3.7). The properness of the action holds also for the set \(\mathcal{S}_g^{\mathrm{a}}\) of archimedean points; but it is important to note that it is not proved yet for the whole space \(\mathcal{S}_g\).
The next step is to define \(\mathcal{C}_g\), the universal Mumford curve over \(\mathbb{Z}\), and then to prove that there exists a proper morphism of analytic spaces over \(\mathbb{Z}\), \(\mathcal{C}_g \rightarrow \mathcal{S}_g\), such that for every \(x \in \mathcal{S}_g\), its preimage in \(\mathcal{C}_g\) is a curve uniformized by the Schottky group \(\Lambda _x\) (Corollary 6.1.3), and so the universal Mumford curve \(\mathcal{C}_g\) can be uniformized by an open subset of the relative projective line over \(\mathcal{S}_g\), \(\mathbb{P}_{\mathcal{S}_g}^1\). The article ends studying the relationship on the non-archimedean Schottky space with the context of the geometric group theory and the theory of moduli spaces of tropical curves (Theorem 6.2.2).
Reviewer: José Javier Etayo (Madrid)Groups of automorphisms of Riemann and Klein surfaces, our joint work with Marston Conderhttps://zbmath.org/1502.140762023-02-24T16:48:17.026759Z"Bujalance, Emilio"https://zbmath.org/authors/?q=ai:bujalance.emilio"Cirre, Francisco-Javier"https://zbmath.org/authors/?q=ai:cirre.francisco-javierMarston Conder made his doctoral dissertation, ``Minimal generating pairs for certain permutation groups'' under the supervision of Graham Higman in 1980. In the same year Emilio Bujalance read his, ``Sobre los subgrupos normales cristalográficos no euclídeos'' (``On non-Euclidean crystallographic normal subgroups''), supervised by Joaquín Arregui. Since then, each of them started a research career on their own as well as making a number of disciples. In 1989, they met for the first time in Groups St. Andrews. Ten years later, the first product of their mathematical collaboration was published in [J. Lond. Math. Soc., II. Ser. 59, No. 2, 573--584 (1999; Zbl 0922.20054)]. A little afterwards, Francisco Javier Cirre, a doctoral student of E. Bujalance (and of José Manuel Gamboa) joined the team. Since then, both E. Bujalance and F. J. Cirre, the authors of the paper under review, have collaborated with M. Conder in the study of groups of automorphisms of Riemann and Klein surfaces.
The present article is a short survey of the main results in that collaboration of the authors and M. Conder. It splits into four Sections (2 to 5), according to the precise questions studied. Section 2 is devoted to the extendability of group actions. This problem consists on deciding whether a subgroup \(G\) of the automorphism group of a surface \(S\) is the full group of automorphisms of \(S\), or else a proper subgroup. This was the problem solved for cyclic groups and Riemann surfaces in the first joint paper of Bujalance and Conder (see above), and extended later to non-cyclic groups, and to bordered or non-orientable Klein surfaces, by the triple Bujalance, Cirre and Conder. In the experience of the reviewer, these results are crucial when determining the automorphism groups of surfaces of a given genus.
In Section 3 the results on automorphism groups of bordered or non-orientable Klein surfaces are described. They include results for surfaces of low genus, or for large order (in terms of the genus) groups, and not only give the abstract group definition, but also classify the actions up to topological equivalence.
Section 4 deals with the relationship between the automorphism group of a Klein surface \(X\), and that of its double cover, which is a Riemann surface \(S\), in order to determine whether \(\mathrm{Aut}(S)\) is \(\mathrm{Aut}(X) \times C_2\), or else \(S\) has additional automorphisms.
Finally, in Section 5 the authors describe the results on the automorphism groups of pseudo-real Riemann surfaces, those admitting anticonformal automorphisms but no anticonformal involutions.
All this joint results by Bujalance, Cirre and Conder arose from multiple visits to Auckland and Madrid, and from a lot of meetings throughout the world. The article is a nice piece of work which gives a useful compilation of key results on the topic, and highlights how human qualities and friendship help to make mathematics.
Reviewer: José Javier Etayo (Madrid)A comparison of generalized opers and \((G, P)\)-opershttps://zbmath.org/1502.140862023-02-24T16:48:17.026759Z"Yang, Mengxue"https://zbmath.org/authors/?q=ai:yang.mengxueSummary: The goal of this paper is to describe the relationship between generalized \(B\)-opers, generalized \(\text{SO}(2n,{\mathbb{C}})\)-opers and \((G, P)\)-opers. In particular, we show that to each generalized \(B\)-oper there is a naturally associated \((G, P)\)-oper, but there are some \((G, P)\)-opers that do not arise as generalized \(B\)-opers or \(\text{SO}(2n,{\mathbb{C}})\)-opers.Schottky presentations of positive representationshttps://zbmath.org/1502.141322023-02-24T16:48:17.026759Z"Burelle, Jean-Philippe"https://zbmath.org/authors/?q=ai:burelle.jean-philippe"Treib, Nicolaus"https://zbmath.org/authors/?q=ai:treib.nicolausThe paper under review deals with higher Teichmüller spaces for an oriented surface \(\Sigma\) of negative Euler characteristic, \(\Gamma = \pi_1(\Sigma)\) its fundamental group, and \(G\) a simple real Lie group. These spaces are open subsets of \(\mathrm{Hom}(\Gamma, G)/\hspace{-1mm}/G\). For closed surfaces the first such spaces were obtained by \textit{N. J. Hitchin} in [Topology 31, No. 3, 449--473 (1992; Zbl 0769.32008)] and are now called Hitchin components. Taking \(G = \text{PSL}(n,\mathbb{R})\), those components were studied by \textit{V. Fock} and \textit{A. Goncharov} in [Publ. Math., Inst. Hautes Étud. Sci. 103, 1--211 (2006; Zbl 1099.14025)] and \textit{F. Labourie} in [Invent. Math. 165, No. 1, 51--114 (2006; Zbl 1103.32007)]. In this last paper, and for surfaces with boundary, the space of positive representations, analog to the Hitchin components, was defined. Later, \textit{M. Burger} et al. studied in [Ann. Math. (2) 172, No. 1, 517--566 (2010; Zbl 1208.32014)] a new family of higher Teichmüller spaces, that of maximal representations, for both bordered and unbordered surfaces. A feature appearing to connect these higher Teichmüller theories is the cyclic structure on the boundary of \(G\). Then, the authors of the present paper defined the concept of a generalized Schottky group of automorphisms of a space admitting a partial cyclic order, and proved that maximal representations for bordered surfaces are an example of it, [\textit{J.-P. Burelle} and \textit{N. Treib}, Geom. Dedicata 195, 215--239 (2018; Zbl 1400.22011)].
In the paper under review, a partial cyclic order on the space of complete oriented flags in \(\mathbb{R}^n\) is defined, and the Fock-Goncharov positive representations into \(\text{PSL}(n,\mathbb{R})\) are shown to be generalized Schottky groups acting on that cyclically ordered space (Theorem 1.1). If the Schottky group is purely hyperbolic, that is to say, the intervals defining it do not share endpoints, then the representation \(\rho : F_g \rightarrow \mathrm{PSL}(n,\mathbb{R})\) is \(B\)-Anosov, for \(F_g\) the free group on \(g\) generators, and \(B\) a Borel subgroup (Theorem 1.3). Then, for even \(n = 2k\), a fundamental domain is built in the projective space \(\mathbb{RP}^{2k-1}\). If the Schottky representation is purely hyperbolic Anosov, the orbit of that fundamental domain is the cocompact domain of discontinuity \(D \subset \mathbb{RP}^{2k-1}\). In Theorem 1.4 the fundamental domain is proved to be bounded by finitely many polynomial hypersurfaces. A similar result is obtained in Theorem 1.5 for \(n = 4k+3\), but then \(D\) is a subset of the sphere \(S^{4k+2}\). No analog result, neither in projective space nor in sphere, holds for \(n = 4k+1\).
Reviewer: José Javier Etayo (Madrid)On the E. Study maps for the dual quaternionshttps://zbmath.org/1502.150172023-02-24T16:48:17.026759Z"Yüce, Salim"https://zbmath.org/authors/?q=ai:yuce.salimSummary: In this study, the generalization of the well-known E. Study map for unit dual quaternion vectors in \(H_{\mathbb{D}}^3\) is obtained, where \(\mathbb{D}\) and \(H_{\mathbb{D}}\) are sets of dual numbers and dual quaternions, respectively.A restricted Magnus property for profinite surface groupshttps://zbmath.org/1502.200192023-02-24T16:48:17.026759Z"Boggi, Marco"https://zbmath.org/authors/?q=ai:boggi.marco"Zalesskii, Pavel"https://zbmath.org/authors/?q=ai:zalesskij.pavel-aSummary: Magnus proved in 1930 that, given two elements \( x\) and \( y\) of a finitely generated free group \( F\) with equal normal closures \( \langle x\rangle ^F=\langle y\rangle ^F\), \( x\) is conjugated either to \( y\) or \( y^{-1}\). More recently, this property, called the Magnus property, has been generalized to oriented surface groups. In this paper, we consider an analogue property for profinite surface groups. While the Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for \( \mathscr {S}\) a class of finite groups, we prove that if \( x\) and \( y\) are \textit{algebraically simple} elements of the pro-\( \mathscr {S}\) completion \( \widehat {\Pi }^{\mathscr {S}}\) of an orientable surface group \( \Pi \) such that, for all \( n\in \mathbb{N}\), there holds \( \langle x^n\rangle ^{\widehat {\Pi }^{\mathscr {S}}}=\langle y^n\rangle ^{\widehat {\Pi }^{\mathscr {S}}}\), then \( x\) is conjugated to \( y^s\) for some \( s\in (\widehat {\mathbb{Z}}^{\mathscr {S}})^\ast \). As a matter of fact, a much more general property is proved and further extended to a wider class of profinite completions.
The most important application of the theory above is a generalization of the description of centralizers of profinite Dehn twists given in [\textit{M. Boggi}, Trans. Am. Math. Soc. 366, No. 10, 5185--5221 (2014; Zbl 1298.14030)] to profinite Dehn multitwists.Geometry of measures in random systems with complete connectionshttps://zbmath.org/1502.280082023-02-24T16:48:17.026759Z"Mihailescu, Eugen"https://zbmath.org/authors/?q=ai:mihailescu.eugen-gh"Urbański, Mariusz"https://zbmath.org/authors/?q=ai:urbanski.mariuszOne can cite authors' description of this research:
``In this paper, we study relations between countable conformal iterated function systems (IFS) with arbitrary overlaps, Smale's endomorphisms, and random systems with complete connections, from the point of view of their geometric and ergodic properties. We provide a common framework for studying measures with certain invariance properties and their dimensions in these systems.
We prove that stationary measures for countable conformal IFS with overlaps and place-dependent probabilities, are exact dimensional; moreover we determine their Hausdorff dimension. Next, we construct a family of fractals in the limit set of a countable IFS with overlaps \(\mathcal{S}\), and study the dimension for certain measures supported on these subfractals. In particular, we obtain families of measures on these subfractals which are related to the geometry of the system \(\mathcal{S}\).''
In the survey of this paper, such notions as conformal iterated function systems, finite iterated function systems with place-dependent probabilities (weights), and random systems with complete connections, as well as finite conformal IFS with overlaps, countable conformal IFS with overlaps, and finite iterated function systems with place-dependent probabilities, etc., are noted.
One can remark that the present research deals with the following notions: countable IFS \(\mathcal{S}\) with place-dependent probabilities and arbitrary overlaps, an arbitrary countable IFS with overlaps \(\mathcal{S}\) which satisfies a condition of pointwise non-accumulation, and random systemswith complete connections, as well as Smale's skew product endomorphism, etc. Also, the notion of finite IFS with place-dependent probabilities is extended to countable iterated function systems with overlaps and place-dependent probabilities.
Reviewer: Symon Serbenyuk (Kyïv)A first course in complex analysishttps://zbmath.org/1502.300012023-02-24T16:48:17.026759Z"Willms, Allan R."https://zbmath.org/authors/?q=ai:willms.allan-rPublisher's description: This book introduces complex analysis and is appropriate for a first course in the subject at typically the third-year University level. It introduces the exponential function very early but does so rigorously. It covers the usual topics of functions, differentiation, analyticity, contour integration, the theorems of Cauchy and their many consequences, Taylor and Laurent series, residue theory, the computation of certain improper real integrals, and a brief introduction to conformal mapping. Throughout the text an emphasis is placed on geometric properties of complex numbers and visualization of complex mappings.A first course in complex analysishttps://zbmath.org/1502.300022023-02-24T16:48:17.026759Z"Willms, Allan R."https://zbmath.org/authors/?q=ai:willms.allan-rFor the original edition see [Zbl 1502.30001].On a family of holomorphic self-maps of the unit diskhttps://zbmath.org/1502.300032023-02-24T16:48:17.026759Z"Ounaies, M."https://zbmath.org/authors/?q=ai:ounaies.myriam"Sac-Épée, J. M."https://zbmath.org/authors/?q=ai:sac-epee.jean-marcSummary: We give a characterization of the sets \(D_p\) (\(1<p<2\)) of complex numbers \(c\) such that \(z\mapsto \frac{1+z}{2}+c\left( \frac{1-z}{2}\right)^p\) is a self-map of the closed unit disk and we show that these sets are increasing with respect to \(p\).Bohr phenomenon for certain close-to-convex analytic functionshttps://zbmath.org/1502.300042023-02-24T16:48:17.026759Z"Allu, Vasudevarao"https://zbmath.org/authors/?q=ai:allu.vasudevarao"Halder, Himadri"https://zbmath.org/authors/?q=ai:halder.himadriSummary: We say that a class \({\mathcal{G}}\) of analytic functions \(f\) of the form \(f(z)=\sum_{n=0}^{\infty } a_nz^n\) in the unit disk \({\mathbb{D}}:=\{z\in{\mathbb{C}}: |z|<1\}\) satisfies a Bohr phenomenon if for the largest radius \(R_f<1\), the following inequality
\[
\sum \limits_{n=1}^{\infty } |a_nz^n| \le d(f(0),\partial f({\mathbb{D}}) )
\]
holds for \(|z|=r\le R_f\) and for all functions \(f \in{\mathcal{G}} \). The largest radius \(R_f\) is called Bohr radius for the class \({\mathcal{G}} \). In this article, we obtain the Bohr radius for certain subclasses of close-to-convex analytic functions. We establish the Bohr phenomenon for certain analytic classes \({\mathcal{S}}_c^*(\phi ),\,{\mathcal{C}}_c(\phi ),\, {\mathcal{C}}_s^*(\phi ),\, {\mathcal{K}}_s(\phi )\) and obtain the radius \(R_f\) such that the Bohr phenomenon for these classes holds for \(|z|=r\le R_f\). As a consequence of these results, we obtain several interesting corollaries about the Bohr phenomenon for the aforesaid classes.Various operators in relation to fractional order calculus and some of their applications to normalized analytic functions in the open unit diskhttps://zbmath.org/1502.300052023-02-24T16:48:17.026759Z"Irmak, Hüseyin"https://zbmath.org/authors/?q=ai:irmak.huseyinSummary: The main object of this scientific work is firstly to introduce various operators of fractional calculus (that is that fractional integral and fractional derivative(s)) in certain domains of the complex plane, then to determine certain results correlating with normalized analytic functions, which are analytic in certain domains in the complex plane, as a few applications of those operators, and also to present a number of extensive implications of them as special results.Zero-free regions for lacunary type polynomialshttps://zbmath.org/1502.300062023-02-24T16:48:17.026759Z"Malik, S. Ahmad"https://zbmath.org/authors/?q=ai:malik.shabir-ahmad"Kumar, A."https://zbmath.org/authors/?q=ai:kumar.arun|kumar.ajay.2|kumar.a-p-siva|kumar.aman|kumar.aviral|kumar.a-r-hemanth|kumar.ashok|kumar.ashwani|kumar.amod|kumar.amrendra|kumar.ashutosh|kumar.abhinav|kumar.a-anil|kumar.a-selva|kumar.arjun|kumar.a-muneesh|kumar.ashisha|kumar.arya|kumar.a-v-senthil|kumar.anoj|kumar.ankan|kumar.anmol|kumar.arpith|kumar.ananya|kumar.ashwin|kumar.adarsh|kumar.a-ramamurthy-vidhya|kumar.arun-n|kumar.a-vincent-antony|kumar.anup|kumar.abhay|kumar.asutosh|kumar.amal|kumar.aounon|kumar.anoop|kumar.a-vincent-anthony|kumar.arunod|kumar.a-vinod|kumar.akansha|kumar.a-s-vinod|kumar.aloke|kumar.apurv|kumar.anant|kumar.atendra|kumar.angamuthu-sathish|kumar.akhil|kumar.a-senthil|kumar.alpesh|kumar.agney|kumar.abhinit|kumar.a-b|kumar.a-g-vijaya|kumar.akshat|kumar.abbay|kumar.avanish|kumar.abhishek|kumar.amit|kumar.amit.1|kumar.akash|kumar.anshuman|kumar.ashwini|kumar.a-satish|kumar.animesh|kumar.anudeep|kumar.anjani|kumar.ambreesh|kumar.ahlad|kumar.anuj|kumar.anil|kumar.awadhesh|kumar.amresh|kumar.athira-satheesh|kumar.aanjaneya|kumar.anish|kumar.amritesh|kumar.anshul|kumar.ankush|kumar.amit.2|kumar.amar|kumar.ashim|kumar.a-sameer|kumar.arvind|kumar.a-vanav|kumar.ajit|kumar.amitesh|kumar.amruth-n|kumar.alok|kumar.ashvini|kumar.ananth-s-r|kumar.awanish|kumar.ashock|kumar.avdhesh|kumar.aishwarya|kumar.ayu|kumar.ajendra|kumar.anand|kumar.awdhesh|kumar.arup|kumar.avneesh|kumar.ajeet|kumar.akshay|kumar.avadhesh|kumar.a-ramesh|kumar.ankit|kumar.akshi|kumar.abhimanyu|kumar.akhilesh|kumar.anshu|kumar.apurva|kumar.amarendra|kumar.abhimanyua|kumar.aditya|kumar.awani|kumar.atul|kumar.anurag|kumar.avinash|kumar.archana|kumar.amioy|kumar.akarsh|kumar.asheel|kumar.abhass|kumar.ashish"Zargar, B. Ahmad"https://zbmath.org/authors/?q=ai:zargar.bashir-ahmadSummary: This paper aims to set an account of zero-free regions for lacunary type polynomials whose coefficients or their real and imaginary parts are subjected to certain restrictions. We also find bounds concerning the number of zeros in a specific annular region.Properties of sine series in weighted spaceshttps://zbmath.org/1502.300072023-02-24T16:48:17.026759Z"Vukolova, T. M."https://zbmath.org/authors/?q=ai:vukolova.t-m"Simonov, B. V."https://zbmath.org/authors/?q=ai:simonov.boris-vSummary: Sums of double sine series with coefficients multiply monotone by subsequences are studied. Conditions under which these sums belong to weighted spaces are obtained.A Gaussian version of Littlewood's theorem for random power serieshttps://zbmath.org/1502.300082023-02-24T16:48:17.026759Z"Cheng, Guozheng"https://zbmath.org/authors/?q=ai:cheng.guozheng"Fang, Xiang"https://zbmath.org/authors/?q=ai:fang.xiang"Guo, Kunyu"https://zbmath.org/authors/?q=ai:guo.kunyu"Liu, Chao"https://zbmath.org/authors/?q=ai:liu.chao.1Summary: We prove a Littlewood-type theorem for random analytic functions associated with not necessarily independent Gaussian processes. We show that if we randomize a function in the Hardy space \(H^2(\mathbb{D})\) by a Gaussian process whose covariance matrix \(K\) induces a bounded operator on \(l^2\), then the resulting random function is almost surely in \(H^p(\mathbb{D})\) for any \(p>0\). The case \(K=\mathrm{Id}\), the identity operator, recovers Littlewood's theorem. A new ingredient in our proof is to recast the membership problem as the boundedness of an operator. This reformulation enables us to use tools in functional analysis and is applicable to other situations.A three-phase elliptical inhomogeneity in nonlinearly coupled thermoelectric materialshttps://zbmath.org/1502.300092023-02-24T16:48:17.026759Z"Wang, Xu"https://zbmath.org/authors/?q=ai:wang.xu"Schiavone, Peter"https://zbmath.org/authors/?q=ai:schiavone.peterSummary: We use complex variable techniques to derive a rigorous closed-form solution to the nonlinearly coupled thermoelectric problem associated with a three-phase elliptical inhomogeneity with two confocal elliptical interfaces when the matrix is subjected to uniform remote electric current density and uniform remote energy flux. Explicit expressions for each of the six analytic functions characterizing the thermoelectric fields in the three phases are obtained. The analysis indicates that the internal temperature exhibits a quadratic distribution within the elliptical inhomogeneity.Asymptotic estimates for analytic functions in strips and their derivativeshttps://zbmath.org/1502.300102023-02-24T16:48:17.026759Z"Beregova, G. I."https://zbmath.org/authors/?q=ai:beregova.g-i"Fedynyak, S. I."https://zbmath.org/authors/?q=ai:fedynyak.stepan-ivanovich"Filevych, P. V."https://zbmath.org/authors/?q=ai:filevych.petro-vSummary: Let \(-\infty\le A_0< A\le +\infty\), \(\Phi\) be a continuous function on \([a,A)\) such that for every \(x\in\mathbb{R}\) we have \(x\sigma-\Phi(\sigma)\to-\infty\) as \(\sigma\uparrow A, \widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma):\sigma\in [a,A)\}\) be the Young-conjugate function of \(\Phi\), \({\Phi}_*(x)=\widetilde{\Phi}(x)/x\) for all sufficiently large \(x\), and \(F\) be an analytic function in the strip \(\{s\in\mathbb{C}\colon A_0<\operatorname{Re}s<A\}\) such that the quantity \(S(\sigma,F)=\sup\{|F(\sigma+it)|\colon t\in\mathbb{R}\}\) is finite for all \(\sigma\in(A_0,A)\) and \(F(s)\not\equiv0\). It is proved that if
\[\ln S(\sigma,F)\le(1+o(1)\Phi(\sigma) \text{ as } \sigma\uparrow A,\]
then
\[ \varlimsup_{\sigma\uparrow A}\frac{S(\sigma,F')}{S(\sigma,F){\Phi}_*^{-1}(\sigma)}\le c_0,\]
where \(c_0<1,1276\) is an absolute constant. From previously obtained results it follows that \(c_0\) cannot be replaced by a constant less than 1.Multipliers on class of Dirichlet series having vector valued frequencieshttps://zbmath.org/1502.300112023-02-24T16:48:17.026759Z"Dua, Nibha"https://zbmath.org/authors/?q=ai:dua.nibha"Kumar, Niraj"https://zbmath.org/authors/?q=ai:kumar.niraj(no abstract)On maximum modulus of polynomials with restricted zeroshttps://zbmath.org/1502.300122023-02-24T16:48:17.026759Z"Chanam, Barchand"https://zbmath.org/authors/?q=ai:chanam.barchand"Devi, Khangembam Babina"https://zbmath.org/authors/?q=ai:devi.khangembam-babina"Krishnadas, Kshetrimayum"https://zbmath.org/authors/?q=ai:krishnadas.kshetrimayum"Devi, Maisnam Triveni"https://zbmath.org/authors/?q=ai:devi.maisnam-triveniSummary: Let \(p(z)\) be a polynomial of degree \(n\) with zero of multiplicity \(s\) at the origin and the remaining zeros in \(|z|\geq k\) or in \(|z|\leq k\), \(k>0\). In this paper, first we obtain inequalities about the dependence of \(|p(Rz)|\) on \(|p(rz)|\), where \(|z|=1\), for \(r^2\leq Rr\leq k^2\) or \(k^2\leq Rr \leq R^2\). Further, another similar inequality for the class of polynomials having all zeros in \(|z|\geq k\), \(k>0\) is also proved for \(0<r\leq R\leq k\). Our results improve as well as generalize certain well-known polynomial inequalities.Remark on some recent inequalities of a polynomial and its derivativeshttps://zbmath.org/1502.300132023-02-24T16:48:17.026759Z"Chanam, Barchand"https://zbmath.org/authors/?q=ai:chanam.barchand"Devi, Khangembam Babina"https://zbmath.org/authors/?q=ai:devi.khangembam-babina"Singh, Thangjam Birkramjit"https://zbmath.org/authors/?q=ai:singh.thangjam-birkramjitSummary: We point out a flaw in a result proved by \textit{G. Singh} and \textit{W. M. Shah} [Kyungpook Math. J. 57, No. 4, 537--543 (2017; Zbl1400.30007)] which was recently published in Kyungpook Mathematical Journal. Further, we point out an error in another result of the same paper which we correct and obtain integral extension of the corrected form.Growth estimates of a polynomial not vanishing in a diskhttps://zbmath.org/1502.300142023-02-24T16:48:17.026759Z"Hussain, Imtiaz"https://zbmath.org/authors/?q=ai:hussain.imtiazSummary: This paper deals with the problem of finding some upper bound estimates for the maximal modulus of a lucunary polynomial on a disk of radius \(R, R\ge 1\) under the assumption that the polynomial does not vanish in another disk with radius \(k\), \(k\ge 1\). The obtained results generalize as well as sharpen some already known estimates due to [\textit{A. Dalal} et al., Anal. Theory Appl. 36, No. 2, 225--234 (2020; Zbl 1474.30005); \textit{K. K. Dewan} et al., J. Interdiscip. Math. 1, No. 2--3, 129--140 (1998; Zbl 0923.30001); \textit{R. B. Gardner} et al., Int. J. Pure Appl. Math. 13, No. 4, 491--498 (2004; Zbl 1054.30009); \textit{R. B. Gardner} et al., J. Inequal. Pure Appl. Math. 6, No. 2, Paper No. 53, 9 p. (2005; Zbl 1081.30006)].A general approach to the study of the convergence of Picard iteration with an application to Halley's method for multiple zeros of analytic functionshttps://zbmath.org/1502.300152023-02-24T16:48:17.026759Z"Ivanov, Stoil I."https://zbmath.org/authors/?q=ai:ivanov.stoil-iSummary: In this paper, we define a new wide class of iteration functions and then we use it to prove a general convergence theorem that provides exact domain of initial approximations to guarantee the high \(Q\)-order of convergence of Picard iteration generated by this class of functions. As an application of this theorem, we prove some local convergence theorems about the famous Halley's method for simple and multiple zeros of analytic functions. All obtained results are supplied with a priori and a posteriori error estimates as well as with assessments of the asymptotic error constants.On the Erdős-Lax inequalityhttps://zbmath.org/1502.300162023-02-24T16:48:17.026759Z"Kumar, Prasanna"https://zbmath.org/authors/?q=ai:kumar.prasanna-v-kSummary: The Erdős-Lax Theorem states that if \(P(z)=\sum_{\nu=1}^n a_{\nu}z^{\nu}\) is a polynomial of degree \(n\) having no zeros in \(|z|<1,\) then
\[
\max_{|z|=1}|P'(z)|\leq\frac{n}{2}\max_{|z|=1}|P(z)|.\tag{1}
\]
In this paper, we prove a sharpening of the above inequality (1). In order to prove our result we prove a sharpened form of the well-known Theorem of Laguerre on polynomials, which itself could be of independent interest.On sharpening and generalization of Rivlin's inequalityhttps://zbmath.org/1502.300172023-02-24T16:48:17.026759Z"Kumar, Prasanna"https://zbmath.org/authors/?q=ai:kumar.prasanna-v-k"Milovanovic, Gradimir V."https://zbmath.org/authors/?q=ai:milovanovic.gradimir-vSummary: An inequality due to \textit{T. J. Rivlin} [Am. Math. Mon. 67, 251--253 (1960; Zbl 0097.00802)] states that if \(P(z)\) is a polynomial of degree \(n\) having no zeros in \(|z|<1\), then
\[
\max\limits_{|z| = r}|P(z)|\geq\left(\frac{1+r}{2}\right)^n\max\limits_{|z| = 1}|P(z)|
\]
for \(0 \leq r \leq 1\). In this paper, we prove some generalizations of the above Rivlin's inequality which sharpens Rivlin's inequality as a special case. Some important consequences of these results are also discussed and some related inequalities are obtained.Inequalities of Turán-type for algebraic polynomialshttps://zbmath.org/1502.300182023-02-24T16:48:17.026759Z"Milovanović, Gradimir V."https://zbmath.org/authors/?q=ai:milovanovic.gradimir-v"Mir, Abdullah"https://zbmath.org/authors/?q=ai:mir.abdullah"Hussain, Adil"https://zbmath.org/authors/?q=ai:hussain.adilSummary: In this paper, we establish some new inequalities in the complex plane that are inspired by some classical Turán-type inequalities that relate the norm of a univariate complex coefficient polynomial and its derivative on the unit disk. The obtained results produce various inequalities in the supremum-norm and in the integral-norm of a polynomial that are sharper than the previous ones while taking into account the placement of the zeros and some of the extremal coefficients of the underlying polynomial. Moreover, our results besides derive polar derivative analogues of some classical Turán-type inequalities also include several interesting generalizations and refinements of some integral inequalities for polynomial as well. Some numerical examples are given in order to graphically illustrate and compare the obtained inequalities with some classical results.Growth estimates of derivatives of a polynomialhttps://zbmath.org/1502.300192023-02-24T16:48:17.026759Z"Mir, Abdullah"https://zbmath.org/authors/?q=ai:mir.abdullah(no abstract)Inequalities of Erdős-Lax-type for a complex polynomialhttps://zbmath.org/1502.300202023-02-24T16:48:17.026759Z"Mir, Abdullah"https://zbmath.org/authors/?q=ai:mir.abdullah"Ahmad, Abrar"https://zbmath.org/authors/?q=ai:ahmad.abrarSummary: If \(P(z)\) is a polynomial of degree \(n\) which does not vanish in \(|z|<k,\,k\leq 1\), then \textit{N. K. Govil} [Proc. Natl. Acad. Sci. India, Sect. A 50, 50--52 (1980; 0493.30003)] proved that
\[
\begin{aligned} \max_{|z|=1}|P'(z)|\leq \frac{n}{1+k^n}\max_{|z|=1}|P(z)|,
\end{aligned}
\]
provided \(|P'(z)|\) and \(|Q'(z)|\) attain maximum at the same point on \(|z|=1\), where \(Q(z)=z^n\overline{P\left(\frac{1}{\overline{z}} \right)}\). In this paper, we prove certain refinements and generalizations of this inequality and related results by taking into account the placement of the zeros and the extremal coefficients of the polynomial.Inequalities concerning rational functions in the complex domainhttps://zbmath.org/1502.300212023-02-24T16:48:17.026759Z"Mir, A."https://zbmath.org/authors/?q=ai:mir.azeem|mir.arnau|mir.abdullah|mir.arshid|mir.aijaz"Hans, S."https://zbmath.org/authors/?q=ai:hans.stephane|hans.sunil|hans.shui-hua|hans.sandeepSummary: We establish some inequalities for the modulus of the derivative of rational functions of Bernstein and Turán type in the sup-norm on the unit disk in the complex plane. These results produce some sharper inequalities while taking into account the placement of zeros of the underlying rational function. Moreover, many inequalities for polynomials and polar derivatives follow as special cases as well.On zeros of the numerator and denominator polynomials of Thiele's continued fractionhttps://zbmath.org/1502.300222023-02-24T16:48:17.026759Z"Pahirya, M. M."https://zbmath.org/authors/?q=ai:pahirya.mykhaylo-m|pagirya.m-mSummary: We prove that the polynomials of canonical numerators and denominators of the interpolation and approximation convergents of Thiele's continued fractions have no common zeros. It is shown that the convergents of Thiele's continued fraction form a staircase sequence of normal Padé approximants. The region containing zeros of the denominator polynomial of the convergent of Thiele's continued fraction is also determined.Integral inequalities for the growth and higher derivative of polynomialshttps://zbmath.org/1502.300232023-02-24T16:48:17.026759Z"Rather, N. A."https://zbmath.org/authors/?q=ai:rather.n-a|rather.nisar-ahmad"Bhat, A."https://zbmath.org/authors/?q=ai:bhatt.abhay-g|bhat.ashoka-k-s|bhat.aijaz-ahmad|bhat.ashwini|bhat.ajaz-ahmad|bhat.ambika|bhat.anha|bhat.adithya|bhat.altaf-ahmad|bhat.akhtar-hussain|bhat.aarif-hussain|bhat.archana"Shafi, M."https://zbmath.org/authors/?q=ai:shafi.mansoor|shafi.mohammadSummary: Let \(P(z)\) be a polynomial of degree \(n\) which does not vanish in \(|z|<1\), it was proved by \textit{S. Gulzar} [Anal. Math. 42, No. 4, 339--352 (2016, Zbl 1374.3001)] that
\[\left|\left|z^sP^{(s)}(z)+\beta\frac{n(n-1)\dots(n-s+1)}{2^s}P(z)\right|\right|_p\leqslant n(n-1)\dots(n-s+1)\left|\left|\left(1+\frac{\beta}{2^s}\right)z+\frac{\beta}{2^s}\right|\right|_p\frac{\left|\left|P(z)\right|\right|_p}{\left|\left|1+z\right|\right|_p}
\] for every \(\beta\in\mathbb{C}\) with \(|\beta|\leqslant 1\), \(1\leqslant s\leqslant n\) and \(0\leqslant p<\infty \). In this paper we extend the above result to the growth of polynomials and also generalize the above and other related results in this direction.On loci of complex polynomials of degree threehttps://zbmath.org/1502.300242023-02-24T16:48:17.026759Z"Sendov, Hristo"https://zbmath.org/authors/?q=ai:sendov.hristo-s"Xiao, Junquan"https://zbmath.org/authors/?q=ai:xiao.junquanSummary: A locus of a complex polynomial is a minimal, with respect to inclusion, set that contains at least one point of every solution of the polarization of the polynomial. The notion of a locus of a complex polynomial was introduced in [\textit{B. Sendov} and \textit{H. Sendov}, Trans. Am. Math. Soc. 366, No. 10, 5155--5184 (2014; Zbl 1298.30005)], where several examples were shown for polynomials of degree three. The study of such objects allows one to improve upon known results about the location of zeros and critical points of complex polynomials, see for example [\textit{B. Sendov} and \textit{H. Sendov}, Math. Proc. Camb. Philos. Soc. 159, No. 2, 253--273 (2015; Zbl 1371.30007)] and [\textit{B. Sendov} and \textit{H. Sendov}, Proc. Am. Math. Soc. 146, No. 8, 3367--3380 (2018; Zbl 1391.30010)]. One of the main results in [\textit{B. Sendov} and \textit{H. Sendov}, Trans. Am. Math. Soc. 366, No. 10, 5155--5184 (2014; Zbl 1298.30005)] is Theorem 5.1, which constructs a locus of a polynomial of degree three inside the smallest disk containing its zeros. The proof of that theorem is very long and relies on intricate geometric constructions. It also relies on a specific positioning of the zeros of the polynomial. The goals of this paper are three-fold. First, we give a simpler and more transparent proof of that result. Then we extend it by constructing a locus holder of the polynomial, when its zeros are in arbitrary position. Finally, third, we answer an open question formulated at the end of Section 7 in [\textit{B. Sendov} and \textit{H. Sendov}, Trans. Am. Math. Soc. 366, No. 10, 5155--5184 (2014; Zbl 1298.30005)]and by doing so discover a new locus of the polynomial \(z^3 + 1\).Inequalities for a polynomial whose zeros are within or outside a given diskhttps://zbmath.org/1502.300252023-02-24T16:48:17.026759Z"Shah, Lubna Wali"https://zbmath.org/authors/?q=ai:wali-shah.lubna"Mir, Mohd Yousf"https://zbmath.org/authors/?q=ai:mir.mohd-yousf"Shah, Wali Mohammad"https://zbmath.org/authors/?q=ai:shah.wali-mohammadSummary: In this paper we prove some results by using a simple but elegant techniques to improve and strengthen some generalizations and refinements of two widely known polynomial inequalities and thereby deduce some useful corollaries.Bernstein-type inequalities preserved by modified Smirnov operatorhttps://zbmath.org/1502.300262023-02-24T16:48:17.026759Z"Shah, Wali Mohammad"https://zbmath.org/authors/?q=ai:shah.wali-mohammad"Bhat, Ishrat Ul Fatima"https://zbmath.org/authors/?q=ai:bhat.ishrat-ul-fatimaSummary: In this paper we consider a modified version of Smirnov operator and obtain some Bernstein-type inequalities preserved by this operator. In particular, we prove some results which in turn provide the compact generalizations of some well-known inequalities for polynomials.System of polynomials of complex variable related to the classical systems of orthogonal polynomialshttps://zbmath.org/1502.300272023-02-24T16:48:17.026759Z"Sukhorolsky, M. A."https://zbmath.org/authors/?q=ai:sukhorolskyi.m-a"Veselovska, O. V."https://zbmath.org/authors/?q=ai:veselovska.olga-v"Dostoina, V. V."https://zbmath.org/authors/?q=ai:dostoina.veronika-vSummary: We study the properties of the systems of polynomials of complex variable represented in the form of contour integrals with kernel functions analytic at infinity. We formulate the conditions for the existence of functions associated with these polynomials and sufficient conditions for the expansion of analytic functions in series in these polynomials. The accumulated results can be used to find the expansions of functions in series in the classical orthogonal polynomials in complex domains, the integral representations for some of these polynomials, the dependences of monomials \(z^n\) of these polynomials, and other relations.On the growth of maximum modulus of rational functions with prescribed poleshttps://zbmath.org/1502.300282023-02-24T16:48:17.026759Z"Wali Shah, Lubna"https://zbmath.org/authors/?q=ai:wali-shah.lubnaSummary: In this paper we prove a sharp growth estimate for rational functions with prescribed poles and restricted zeros in the Chebyshev norm on the unit disk in the complex domain. In particular we extend a polynomial inequality due to \textit{V. N. Dubinin} [Zap. Nauchn. Semin. POMI 337, 101--112 (2006); translation in J. Math. Sci., New York 143, No. 3, 3069--3076 (2007; Zbl 1117.30016)] to rational functions which also improves a result of \textit{N. K. Govil} and \textit{R. N. Mohapatra} [in: Approximation theory. In memory of A. K. Varma. New York, NY: Marcel Dekker. 255--263 (1998; Zbl 0902.41011)]Generalization of some inequalities to the class of generalized derivativehttps://zbmath.org/1502.300292023-02-24T16:48:17.026759Z"Wani, Irfan Ahmad"https://zbmath.org/authors/?q=ai:wani.irfan-ahmad"Mir, Mohammad Ibrahim"https://zbmath.org/authors/?q=ai:mir.mohammad-ibrahim"Nazir, Ishfaq"https://zbmath.org/authors/?q=ai:nazir.ishfaqSummary: In this paper, we obtain some inequalities concerning the class of generalized derivative and generalized polar derivative which are analogous respectively to the ordinary derivative and polar derivative of polynomials.Polinomios estableshttps://zbmath.org/1502.300302023-02-24T16:48:17.026759Z"Gasull, Armengol"https://zbmath.org/authors/?q=ai:gasull.armengol(no abstract)Level sets of potential functions bisecting unbounded quadrilateralshttps://zbmath.org/1502.300312023-02-24T16:48:17.026759Z"Nasser, Mohamed M. S."https://zbmath.org/authors/?q=ai:nasser.mohamed-m-s"Nasyrov, Semen"https://zbmath.org/authors/?q=ai:nasyrov.semen-r"Vuorinen, Matti"https://zbmath.org/authors/?q=ai:vuorinen.matti-kSummary: We study the mixed Dirichlet-Neumann problem for the Laplace equation in the complement of a bounded convex polygonal quadrilateral in the extended complex plane. The Dirichlet / Neumann conditions at opposite pairs of sides are \(\{0, 1\}\) and \(\{0, 0\}\), resp. The solution to this problem is a harmonic function in the unbounded complement of the polygon known as the \textit{potential function} of the quadrilateral. We compute the values of the potential function \(u\) including its value at infinity. The main result of this paper is Theorem 4.3 which yields a formula for \(u(\infty)\) expressed in terms of the angles of the polygonal given quadrilateral and the well-known special functions. We use two independent numerical methods to illustrate our result. The first method is a Mathematica program and the second one is based on using the MATLAB toolbox PlgCirMap. The case of a quadrilateral, which is the exterior of the unit disc with four fixed points on its boundary, is considered as well.Characterizations of convergence by a given set of angles in simply connected domainshttps://zbmath.org/1502.300322023-02-24T16:48:17.026759Z"Zarvalis, Konstantinos"https://zbmath.org/authors/?q=ai:zarvalis.konstantinosSummary: Let \(\Delta\) be a simply connected domain and \(f:\mathbb{D}\to\Delta\), where \(\mathbb{D}\) is the unit disk, be a corresponding Riemann map. Let \(\{z_n\}\subset\Delta\) be a sequence with no accumulation points inside \(\Delta\). In the present article, we give necessary and sufficient conditions in terms of hyperbolic geometry which certify that \(\{f^{-1}(z_n)\}\) converges to a point of \(\partial\mathbb{D}\) by a certain angle \(\theta\) or by a certain set of angles \([\theta_1, \theta_2]\).On a class of analytic multivalent functions in \(q\)-analogue associated with lemniscate of Bernoullihttps://zbmath.org/1502.300332023-02-24T16:48:17.026759Z"Ahmad, Bakhtiar"https://zbmath.org/authors/?q=ai:ahmad.bakhtiar"Darus, Maslina"https://zbmath.org/authors/?q=ai:darus.maslina"Khan, Nasir"https://zbmath.org/authors/?q=ai:khan.nasir|khan.nasir-saeed|khan.nasir-m"Khan, Raees"https://zbmath.org/authors/?q=ai:khan.raees"Khan, Muhammad Ghaffar"https://zbmath.org/authors/?q=ai:khan.muhammad-ghaffarSummary: The object of the paper is to examine some various interseting properties of analytic multivalent functions in \(q\)-analogue associated with the lemniscate of Bernoulli.Estimates for \(\lambda \)-spirallike functions of complex order on the boundaryhttps://zbmath.org/1502.300342023-02-24T16:48:17.026759Z"Akyel, T."https://zbmath.org/authors/?q=ai:akyel.tugbaSummary: We give some results obtained for \(\lambda \)-spirallike functions of complex order on the boundary of the unit disc \(U\). The sharpness of these results is also proved. Furthermore, three examples of our results are considered.Starlike functions associated with a petal shaped domainhttps://zbmath.org/1502.300352023-02-24T16:48:17.026759Z"Arora, Kush"https://zbmath.org/authors/?q=ai:arora.kush"Kumar, S. Sivaprasad"https://zbmath.org/authors/?q=ai:kumar.shanmugam-sivaprasadSummary: In this paper, we establish some radius results and inclusion relations for starlike functions associated with a petal-shaped domain.On estimates of some coefficient functionals for certain meromorphic univalent functionshttps://zbmath.org/1502.300362023-02-24T16:48:17.026759Z"Bhowmik, Bappaditya"https://zbmath.org/authors/?q=ai:bhowmik.bappaditya"Parveen, Firdoshi"https://zbmath.org/authors/?q=ai:parveen.firdoshiSummary: Let \({\mathcal{V}}_p(\lambda)\) be the class of all functions \(f\) defined on the open unit disc \({{\mathbb{D}}}\) of the complex plane having simple pole at \(z=p\), \(p\in (0,1)\) and analytic in \({{\mathbb{D}}}{\setminus}\{p\}\) satisfying the normalizations \(f(0)=0=f'(0)-1\) such that \(\left| (z/f(z))^2 f^\prime(z)-1\right| < \lambda\) for \(z\in{{\mathbb{D}}}, \lambda \in (0,1]\). In this article, we obtain sharp bounds of the Zalcman and the generalized Zalcman functionals for functions in \({\mathcal{V}}_p(\lambda)\) for some indices of these functionals. As consequences of the obtained results, we pose the Zalcman-like coefficient conjectures for this class of functions. In addition, we estimate bound for the generalised Fekete-Szegö functional along with bounds of the second- and the third-order Hankel determinants for this class of functions.Properties of meromorphic spiral-like functions associated with symmetric functionshttps://zbmath.org/1502.300372023-02-24T16:48:17.026759Z"Breaz, Daniel"https://zbmath.org/authors/?q=ai:breaz.daniel-v"Cotîrlă, Luminița-Ioana"https://zbmath.org/authors/?q=ai:cotirla.luminita-ioana"Umadevi, Elangho"https://zbmath.org/authors/?q=ai:umadevi.elangho"Karthikeyan, Kadhavoor R."https://zbmath.org/authors/?q=ai:karthikeyan.kadhavoor-ragavan(no abstract)Some subclasses of analytic functions involving certain integral operatorhttps://zbmath.org/1502.300382023-02-24T16:48:17.026759Z"Bukhari, Syed Zakar Hussain"https://zbmath.org/authors/?q=ai:bukhari.syed-zakar-hussain"Noor, Khalida Inayat"https://zbmath.org/authors/?q=ai:noor.khalida-inayatSummary: In this paper, we introduce and investigate two new subclasses of analytic functions with bounded boundary and bounded radius rotations by using a certain \(p\)-valent operator which complies with the known Carlson-Shaffer operator for \(p=1\). Both of these operators are heavily explored and have various applications. We investigate some inclusions results and integral preserving properties. We also extend the Ruscheweyh and Sheil-Small convolution preserving properties in the context of these classes. We relate our finding with the existing known results found in the literature regarding this subject.Fekete-Szegö inequalities for a new general subclass of analytic functions involving the \((p,q)\)-derivative operatorhttps://zbmath.org/1502.300392023-02-24T16:48:17.026759Z"Bulut, Serap"https://zbmath.org/authors/?q=ai:bulut.serapSummary: In this work, we introduce a new subclass of analytic functions of complex order involving the \((p,q)\)-derivative operator defined in the open unit disc. For this class, several Fekete-Szegö type coefficient inequalities are derived. We obtain the results of \textit{H. M. Srivastava} et al. [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 113, No. 4, 3563--3584 (2019; Zbl 1427.30032)] as consequences of the main theorem in this study.Univalence criteria and quasiconformal extension of a general integral operatorhttps://zbmath.org/1502.300402023-02-24T16:48:17.026759Z"Deniz, E."https://zbmath.org/authors/?q=ai:deniz.erhan"Kanas, S."https://zbmath.org/authors/?q=ai:kanas.stanislawa-r"Orhan, H."https://zbmath.org/authors/?q=ai:orhan.halitSummary: We present sufficient conditions for the analyticity and univalence for functions defined by an integral operator. Then we refine the result to a quasiconformal extension criterion with the help of Becker's method. Further, new univalence criteria and significant relationships with other available results are given. A number of known univalence conditions follow if the parameters involved in the main results are specialized.On partial sums of four parametric Wright functionhttps://zbmath.org/1502.300412023-02-24T16:48:17.026759Z"Din, Muhey U."https://zbmath.org/authors/?q=ai:din.muhey-uSummary: Special functions and Geometric function theory are close related to each other due to the surprise use of hypergeometric function in the solution of the Bieberbach conjecture. The purpose of this paper is to provide a set of sufficient conditions under which the normalized four parametric Wright function has lower bounds for the ratios to its partial sums and as well as for their derivatives. The sufficient conditions are also obtained by using Alexander transform. The results of this paper are generalized and also improved the work of the author et al. [Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar. 80, No. 2, 79--90 (2018; Zbl 1424.30039)]. Some examples are also discussed for the sake of better understanding of this article.Fekete-Szegö problem for univalent mappings in one and higher dimensionshttps://zbmath.org/1502.300422023-02-24T16:48:17.026759Z"Hamada, Hidetaka"https://zbmath.org/authors/?q=ai:hamada.hidetaka"Kohr, Gabriela"https://zbmath.org/authors/?q=ai:kohr.gabriela"Kohr, Mirela"https://zbmath.org/authors/?q=ai:kohr.mirelaSummary: In this paper, we will give the Fekete-Szegö inequality for the mappings \(f\) in various subclasses of normalized univalent mappings which are the first elements of \(g\)-Loewner chains on the unit disc \(\mathbb{U}\) in \(\mathbb{C}\) and also on the unit ball \(\mathbb{B}\) of a complex Banach space. As an application, we give the estimation of the third coefficient for \(f\) under the condition that the second coefficient of \(f\) is zero. This result gives a generalization of the estimation of the third coefficient for odd univalent functions on the unit disc \(\mathbb{U}\). We also give the Fekete-Szegö inequality for the images of the first elements of \(g\)-Loewner chains on \(\mathbb{U}\) under the Roper-Suffridge type extension operators.Coefficient bounds for a subclass of \(m\)-fold symmetric \(\lambda\)-pseudo bi-starlike functionshttps://zbmath.org/1502.300432023-02-24T16:48:17.026759Z"Jahangiri, Jay M."https://zbmath.org/authors/?q=ai:jahangiri.jay-m"Murugusundaramoorthy, G."https://zbmath.org/authors/?q=ai:murugusundaramoorthy.gangadharan"Vijaya, K."https://zbmath.org/authors/?q=ai:vijaya.kalippan|vijaya.kaliappan|vijaya.kaliyappan"Uma, K."https://zbmath.org/authors/?q=ai:uma.kaliappan|uma.kaliyappan|uma.kalieppanSummary: In this paper we consider a class of \(\lambda\)-pseudo bi-starlike functions defined by subordination and determine the upper bounds for the first two coefficients of \(m\)-fold symmetric functions in this class. We also determine upper bounds for the Fekete-Szegö coefficients of such \(m\)-fold symmetric functions. Our findings for certain cases improve some of the previously published results.
For the entire collection see [Zbl 1410.16001].Coefficient bounds of bi-univalent functions using Faber polynomialhttps://zbmath.org/1502.300442023-02-24T16:48:17.026759Z"Janani, T."https://zbmath.org/authors/?q=ai:janani.thambidurai"Yalcin, S."https://zbmath.org/authors/?q=ai:yalcin.senay|yalcin.seth|yalcin.sibel|yalcin.soydan|yalcin.serife-nurSummary: In this research article, we study a bi-univalent subclass \(\Sigma\) related with Faber polynomial and investigate the coefficient estimate \(|a_n|\) for functions in the considered subclass with a gap series condition. Also, we obtain the initial two coefficient estimates \(|a_2|\), \(|a_3|\) and find the Fekete-Szegö functional \(|a_3-a_2^2|\) for the considered subclass. New results which are further examined are also pointed out in this article.
For the entire collection see [Zbl 1410.16001].Initial estimates of seven coefficients for a subclass of bi-starlike functionshttps://zbmath.org/1502.300452023-02-24T16:48:17.026759Z"Janani, T."https://zbmath.org/authors/?q=ai:janani.thambidurai"Yalçın, S."https://zbmath.org/authors/?q=ai:yalcin.senay|yalcin.soydan|yalcin.sibel|yalcin.seth|yalcin.serife-nurSummary: We study a subclass of bi-starlike functions and obtain, for the first time, the initial seven Taylor-Maclaurin coefficient estimates \(|a_2 |, |a_3|,\dots, |a_7|\) for functions from a subclass of the function class \(\Sigma \). Some new or known consequences of the accumulated results are also pointed out.A new subclass of harmonic univalent functions defined by differential subordinationhttps://zbmath.org/1502.300462023-02-24T16:48:17.026759Z"Joshi, Sayali S."https://zbmath.org/authors/?q=ai:joshi.sayali-s"Joshi, Santosh B."https://zbmath.org/authors/?q=ai:joshi.santosh-bhaurao"Pawar, Haridas"https://zbmath.org/authors/?q=ai:pawar.haridas-hSummary: In this paper, a new subclass of harmonic functions \(SH^0_\delta(n,A,B,\alpha)\) in \(U\) is defined using differential subordination and other properties like coefficient bounds, distortion theorem, radii of starlikeness and convexity, compactness are obtained.On bounds of Toeplitz determinants for a subclass of analytic functionshttps://zbmath.org/1502.300472023-02-24T16:48:17.026759Z"Kamali, Muhammet"https://zbmath.org/authors/?q=ai:kamali.muhammet"Riskulova, Alina"https://zbmath.org/authors/?q=ai:riskulova.alina(no abstract)Starlikeness associated with crescent-shaped regionhttps://zbmath.org/1502.300482023-02-24T16:48:17.026759Z"Kanaga, R."https://zbmath.org/authors/?q=ai:kanaga.r"Ravichandran, V."https://zbmath.org/authors/?q=ai:ravichandran.vSummary: In this paper, we investigate the sufficient conditions for an analytic function \(f\) defined in the open unit disk \(\mathbb{D}\subseteq\mathbb{C}\) satisfying \(f(0)=f^\prime(0)-1= 0\), such that the function \(f\) satisfies
\[
\left|\left(\frac{zf^\prime(z)}{f(z)}\right)^2-1\right| < 2\left|\frac{zf^\prime(z)}{f(z)}\right|,\quad (z\in\mathbb{D}).
\]
Let \(\mathcal{H}(\mathbb{D})\) denote the class of analytic functions defined in \(\mathbb{D}\) and let \(\mathcal{H}[1, n] := \{p_n\in\mathcal{H}(\mathbb{D}): p_n(z) = 1+a_nz^n+a_{n+1}z^{n+1}+\dots\}\), \(n\in\mathbb{N}\). We derive the admissibility conditions for the function \(q(z) := z + \sqrt{1+z^2}\), which maps \(\mathbb{D}\) to the crescent-shaped region and use it to establish the result if the function \(p_n\in\mathcal{H}[1, n]\) with \(\operatorname{Re}p_n(z) > \sqrt{2}-1\) satisfies the second ordered differential condition \(|z^2p_n^{\prime\prime}(z)/p_n(z)| < (n^2(1+\sqrt{2})-2n)/2\sqrt{2}\), (\(z\in\mathbb{D}\)) then \(p_n\prec q\). We also find several first order differential sufficient conditions for the function \(p_n\) to obey the subordination \(p_n\prec q\). Moreover, we provide examples to demonstrate the existence of the functions \(p_n\in\mathcal{H}[1,n]\) to satisfy these sufficient conditions.On \((p, q)\)-quantum calculus involving quasi-subordinationhttps://zbmath.org/1502.300492023-02-24T16:48:17.026759Z"Kavitha, S."https://zbmath.org/authors/?q=ai:kavitha.seetharaman|kavitha.s-ruth-julie"Cho, Nak Eun"https://zbmath.org/authors/?q=ai:cho.nak-eun"Murugusundaramoorthy, G."https://zbmath.org/authors/?q=ai:murugusundaramoorthy.gangadharanSummary: Let \((p, q) \in (0, 1)\). Let the function \(f\) be analytic in \(|z| < 1\). Further, let the \((p, q)\) be differential operator defined as \(\partial_{p,q} f (z) = \frac{f (pz) - f ({qz})} {z ({p - q})}\), \(|z|<1\). In the current investigation, the authors apply the \((p, q)\)-differential operator for few subclasses of univalent functions defined by quasi-subordination. Initial coefficient bounds for the defined new classes are obtained.
For the entire collection see [Zbl 1410.16001].Estimates for analytic functions associated with Schwarz lemma on the boundaryhttps://zbmath.org/1502.300502023-02-24T16:48:17.026759Z"Kaynakkan, Ayşan"https://zbmath.org/authors/?q=ai:kaynakkan.aysan"Örnek, Bülent Nafi"https://zbmath.org/authors/?q=ai:ornek.bulent-nafiSummary: In this paper, we will introduce the class of analytic functions called \(\mathcal{R}\left(\alpha,\lambda \right)\) and explore the different 5 properties of the functions belonging to this class.Differential inequalities associated with Carathéodory functionshttps://zbmath.org/1502.300512023-02-24T16:48:17.026759Z"Kim, In Hwa"https://zbmath.org/authors/?q=ai:kim.in-hwa"Cho, Nak Eun"https://zbmath.org/authors/?q=ai:cho.nak-eunSummary: The purpose of the present paper is to estimate some real parts for certain analytic functions with some applications in connection with certain integral operators and geometric properties. Also we extend some known results as special cases of main results presented here.The sharp bound of the third Hankel determinant for starlike functionshttps://zbmath.org/1502.300522023-02-24T16:48:17.026759Z"Kowalczyk, Bogumiła"https://zbmath.org/authors/?q=ai:kowalczyk.bogumila"Lecko, Adam"https://zbmath.org/authors/?q=ai:lecko.adam"Thomas, Derek K."https://zbmath.org/authors/?q=ai:thomas.derek-keithSummary: In this paper, we prove the sharp inequality \(\vert H_{3,1}(f)\vert\leq 4/9\) for starlike functions \(f\), where \(H_{3,1}(f)\) is the third Hankel determinant, thus solving a long-standing problem.Hankel determinant of certain orders for some subclasses of holomorphic functionshttps://zbmath.org/1502.300532023-02-24T16:48:17.026759Z"Krishna, D. Vamshee"https://zbmath.org/authors/?q=ai:vamshee-krishna.d"Shalini, D."https://zbmath.org/authors/?q=ai:shalini.dSummary: In this paper, we are introducing certain subfamilies of holomorphic functions and making an attempt to obtain an upper bound (UB) to the second and third order Hankel determinants by applying certain lemmas, Toeplitz determinants, for the normalized analytic functions belong to these classes, defined on the open unit disc in the complex plane. For one of the inequality, we have obtained sharp bound.The Booth lemniscate starlikeness radius for Janowski starlike functionshttps://zbmath.org/1502.300542023-02-24T16:48:17.026759Z"Malik, Somya"https://zbmath.org/authors/?q=ai:malik.somya"Ali, Rosihan M."https://zbmath.org/authors/?q=ai:ali.rosihan-mohamed"Ravichandran, V."https://zbmath.org/authors/?q=ai:ravichandran.vSummary: The function \(G_\alpha (z)=1+ z/(1-\alpha z^2)\), \(0\le \alpha <1\), maps the open unit disk \(\mathbb{D}\) onto the interior of a domain known as the Booth lemniscate. Associated with this function \(G_\alpha\) is the recently introduced class \(\mathcal{BS}(\alpha)\) consisting of normalized analytic functions \(f\) on \(\mathbb{D}\) satisfying the subordination \(zf'(z)/f(z) \prec G_\alpha (z)\). Of interest is its connection with known classes \(\mathcal{M}\) of functions in the sense \(g(z)=(1/r)f(rz)\) belongs to \(\mathcal{BS}(\alpha)\) for some \(r\) in (0, 1) and all \(f \in \mathcal{M} \). We find the largest radius \(r\) for different classes \(\mathcal{M} \), particularly when \(\mathcal{M}\) is the class of starlike functions of order \(\beta \), or the Janowski class of starlike functions. As a primary tool for this purpose, we find the radius of the largest disk contained in \(G_\alpha (\mathbb{D})\) and centered at a certain point \(a \in \mathbb{R} \).A new subclass of uniformly convex functions defined by linear operatorhttps://zbmath.org/1502.300552023-02-24T16:48:17.026759Z"Murthy, A. Narasimha"https://zbmath.org/authors/?q=ai:murthy.a-narasimha"Reddy, P. Thirupathi"https://zbmath.org/authors/?q=ai:reddy.pinninti-thirupathi"Niranjan, H."https://zbmath.org/authors/?q=ai:niranjan.hSummary: In this paper, we define a new subclass of uniformly convex functions with negative coefficients and obtain coefficient estimates, extreme points, closure and inclusion theorems, and the radii of starlikeness and convexity for the new subclass. Furthermore, results on partial sums are discussed.
For the entire collection see [Zbl 1410.16001].Holder's inequalities for analytic functions defined by Ruscheweyh-type \(q\)-difference operatorhttps://zbmath.org/1502.300562023-02-24T16:48:17.026759Z"Mustafa, N."https://zbmath.org/authors/?q=ai:mustafa.nazahah|mustafa.naeem|mustafa.nizami|mustafa.noreen|mustafa.nagat-m|mustafa.nabil-hassan"Vijaya, K."https://zbmath.org/authors/?q=ai:vijaya.kaliappan|vijaya.kaliyappan|vijaya.kalippan"Thilagavathi, K."https://zbmath.org/authors/?q=ai:thilagavathi.kandasamy|thilagavathi.k-t"Uma, K."https://zbmath.org/authors/?q=ai:uma.k-p|uma.kalieppan|uma.kaliyappan|uma.kaliappan|uma.kaliyapanSummary: In this paper, we introduce a new generalized class of analytic functions based on Ruscheweyh-type \(q\)-difference operator. We obtain coefficient estimates, Holder's inequality result, and integral means results for \(f\in \mathscr{TJ}_\mu^\eta (\alpha ,\beta ,\gamma ,A,B)\).
For the entire collection see [Zbl 1410.16001].Hermitian Toeplitz determinants for the class \(\mathcal{S}\) of univalent functionshttps://zbmath.org/1502.300572023-02-24T16:48:17.026759Z"Obradović, Milutin"https://zbmath.org/authors/?q=ai:obradovic.milutin"Tuneski, Nicola"https://zbmath.org/authors/?q=ai:tuneski.nikolaSummary: Introducing a new method, we give sharp estimates of the Hermitian Toeplitz determinants of third order for the class \(\mathcal{S}\) of functions univalent in the unit disc. The new approach is also illustrated on some subclasses of the class \(\mathcal{S} \).On certain properties of a univalent function associated with beta functionhttps://zbmath.org/1502.300582023-02-24T16:48:17.026759Z"Oluwayemi, Matthew Olanrewaju"https://zbmath.org/authors/?q=ai:oluwayemi.matthew-olanrewaju"Olatunji, Sunday Olufemi"https://zbmath.org/authors/?q=ai:olatunji.sunday-olufemi"Ogunlade, Temitope O."https://zbmath.org/authors/?q=ai:ogunlade.temitope-olu(no abstract)Univalence criteria for analytic functions obtained using fuzzy differential subordinationshttps://zbmath.org/1502.300592023-02-24T16:48:17.026759Z"Oros, Georgia Irina"https://zbmath.org/authors/?q=ai:oros.georgia-irinaSummary: Ever since \textit{L. A. Zadeh} published the paper [Inf. Control 8, 338--353 (1965; Zbl 0139.24606)] setting the basis of a new theory named fuzzy sets theory, many scientists have developed this theory and its applications. Mathematicians were especially interested in extending classical mathematical results in the fuzzy context. Such an extension was also done relating fuzzy sets theory and geometric theory of analytic functions. The study begun in 2011 has many interesting published outcomes and the present paper follows the line of the previous research in the field. The aim of the paper is to give some references related to the connections already made between fuzzy sets theory and geometric theory of analytic functions and to present some new results that might prove interesting for mathematicians willing to enlarge their views on certain aspects of the merge between the two theories. Using the notions of fuzzy differential subordination and the classical notion of differential subordination for analytic functions, two criteria for the univalence of the analytic functions are stated in this work.On a generalized class of analytic functions related to Bazilevič functionshttps://zbmath.org/1502.300602023-02-24T16:48:17.026759Z"Parida, Laxmipriya"https://zbmath.org/authors/?q=ai:parida.laxmipriya"Bulboacă, Teodor"https://zbmath.org/authors/?q=ai:bulboaca.teodor"Sahoo, Ashok Kumar"https://zbmath.org/authors/?q=ai:sahoo.ashok-kumarSummary: Using operator \(L_p(a, c)\) introduced by \textit{H. Saitoh} [Math. Japon. 44, No. 1, 31--38 (1996; Zbl 0887.30021)] we define the subclass \(H_{p,n}^{ \nu,\mu } (a, c; \phi )\) of the class \(\mathcal{A}(p, n)\) and establish containment, subordination and coefficient inequalities of this subclass. We indicate the connections of our results with earlier results obtained by other researchers.Convexity of polynomials using positivity of trigonometric sumshttps://zbmath.org/1502.300612023-02-24T16:48:17.026759Z"Sangal, Priyanka"https://zbmath.org/authors/?q=ai:sangal.priyanka"Swaminathan, A."https://zbmath.org/authors/?q=ai:swaminathan.ashvin-anand|swaminathan.anbhu|swaminathan.ayyarasu|swaminathan.anand|swaminathan.ashok|swaminathan.adithSummary: Positivity of trigonometric polynomials is of interest for more than a century because of its applications. In this work, we use positivity of trigonometric sine and cosine sums to find the convexity of a polynomial \(f(z)=\displaystyle \sum_{k=1}^n a_kz^k\). Further, we also investigate the radius of convexity \(r\) such that \(f(\mathbb{D}_{\rho})\) is convex where \(\mathbb{D}_{\rho}=\{z;|z|\leq \rho ,\, 0<\rho <1\}\).
For the entire collection see [Zbl 1410.16001].The second Hankel determinant for starlike and convex functions of order alphahttps://zbmath.org/1502.300622023-02-24T16:48:17.026759Z"Sim, Young Jae"https://zbmath.org/authors/?q=ai:sim.youngjae"Thomas, Derek K."https://zbmath.org/authors/?q=ai:thomas.derek-keith"Zaprawa, Paweł"https://zbmath.org/authors/?q=ai:zaprawa.pawelSummary: In recent years, the study of Hankel determinants for various subclasses of normalised univalent functions \(f \in \mathcal{S}\) given by \(f(z)=z+\sum_{n=2}^\infty a_n z^n\) for \(\mathbb{D}=\{z \in \mathbb{C}\,:\,|z|<1\}\) has produced many interesting results. The main focus of interest has been estimating the second Hankel determinant of the form \(H_{2,2}(f)=a_2a_4 - a_3^2\). A non-sharp bound for \(H_{2,2}(f)\) when \(f \in \mathcal{K}(\alpha)\), \(\alpha \in [0,1)\) consisting of convex functions of order \(\alpha\) was found by \textit{D. V. Krishna} and \textit{T. Ramreddy} [Tbil. Math. J. 5, No. 1, 65--76 (2012; Zbl 1279.30017)], and later improved by the second author et al. [Univalent functions. A primer. Berlin: De Gruyter (2018; Zbl 1397.30002)]. In this paper, we give the sharp result. Moreover, we obtain sharp results for \(H_{2,2}(f^{-1})\) for the inverse functions \(f^{-1}\) when \(f \in \mathcal{K}(\alpha)\), and when \(f \in \mathcal{S}^\ast(\alpha)\), the class of starlike functions of order \(\alpha\). Thus, the results in this paper complete the set of problems for the second Hankel determinants of \(f\) and \(f^{-1}\) for the classes \(\mathcal{S}^\ast(\alpha)\), \(\mathcal{K}(\alpha)\), \(\mathcal{S}_\beta^\ast\) and \(\mathcal{K}_\beta\), where \(\mathcal{S}_\beta^\ast\) and \(\mathcal{K}_\beta\) are, respectively, the classes of strongly starlike, and strongly convex functions of order \(\beta \).Third Hankel determinant for a subclass of close-to-star functions associated with exponential functionhttps://zbmath.org/1502.300632023-02-24T16:48:17.026759Z"Singh, Gagandeep"https://zbmath.org/authors/?q=ai:singh.gagandeep"Singh, Gurcharanjit"https://zbmath.org/authors/?q=ai:singh.gurcharanjit(no abstract)Subclasses of analytic functions defined with generalized Salagean operatorhttps://zbmath.org/1502.300642023-02-24T16:48:17.026759Z"Singh, Gurmeet"https://zbmath.org/authors/?q=ai:singh.gurmeet"Singh, Gagandeep"https://zbmath.org/authors/?q=ai:singh.gagandeep"Singh, Gurcharanjit"https://zbmath.org/authors/?q=ai:singh.gurcharanjitSummary: The present investigation deals with certain subclasses of analytic-univalent functions in the open unit disc \(E=\{z:|z|<1\}\). The coefficient estimates, distortion theorem, argument theorem and relation of these classes with some other classes have been studied and the results so obtained generalize the results of several earlier works.Coefficient functionals for alpha-convex functions associated with the exponential functionhttps://zbmath.org/1502.300652023-02-24T16:48:17.026759Z"Śmiarowska, Barbara"https://zbmath.org/authors/?q=ai:smiarowska.barbaraSummary: Let \(\mathcal{A}\) be the class of all normalized analytic functions \(f\) in the unit disk \(\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}\), given by \(f(z)=z+\sum_{n=2}^{\infty}a_n z^n\) for \(z\in\mathbb{D}\). We give the sharp bound for the modulus of the functional \(a_2 a_3 -a_4\), and the second Hankel determinant \(H_{2,2}(f)=a_2 a_4 -a_3^2\) when \(f\in\mathcal{M}_{\alpha} (\exp)\subset\mathcal{A}\), the class of \(\alpha\)-convex functions \((0\leq \alpha \leq 1)\), associated with the exponential function.Some properties of certain class of uniformly convex functions defined by Bessel functionshttps://zbmath.org/1502.300662023-02-24T16:48:17.026759Z"Srinivas, V."https://zbmath.org/authors/?q=ai:srinivas.vasuvedan|srinivas.virinchi|srinivas.vasudevan"Reddy, P. Thirupathi"https://zbmath.org/authors/?q=ai:reddy.pinninti-thirupathi"Niranjan, H."https://zbmath.org/authors/?q=ai:niranjan.hSummary: The aim of the present paper is to investigate some characterization for generalized Bessel functions of first kind that is to be subclass of analytic functions. Furthermore, we studied coefficient estimates, radius of< starlikeness, convexity, close-to-convexity, and convex linear combinations for the class \(UB (\gamma, k, c)\). Finally we proved integral means inequalities for the class.
For the entire collection see [Zbl 1410.16001].Some sharp results on coefficient estimate problems for four-leaf-type bounded turning functionshttps://zbmath.org/1502.300672023-02-24T16:48:17.026759Z"Sunthrayuth, Pongsakorn"https://zbmath.org/authors/?q=ai:sunthrayuth.pongsakorn"Jawarneh, Yousef"https://zbmath.org/authors/?q=ai:jawarneh.yousef"Naeem, Muhammad"https://zbmath.org/authors/?q=ai:naeem.muhammad-abubakr|naeem.muhammad-nawaz"Iqbal, Naveed"https://zbmath.org/authors/?q=ai:iqbal.naveed-h"Kafle, Jeevan"https://zbmath.org/authors/?q=ai:kafle.jeevan(no abstract)Subordination-implication problems concerning the nephroid starlikeness of analytic functionshttps://zbmath.org/1502.300682023-02-24T16:48:17.026759Z"Swaminathan, Anbhu"https://zbmath.org/authors/?q=ai:swaminathan.anbhu"Wani, Lateef Ahmad"https://zbmath.org/authors/?q=ai:wani.lateef-ahmadSummary: Let \(\mathcal{A}\) be the set of all analytic functions \(f\) defined on the open unit disk \(\mathbb{D}\) satisfying \(f(0)=f'(0)-1=0\). Let \(\varphi_{Ne}(z):=1+z-z^3=3\) be the recently introduced Carathéodory function which maps the unit circle \(\partial\mathbb{D}\) onto a 2-cusped kidney-shaped curve called \textit{nephroid} given by \(\left((u-1)^2+v^2-\frac{4}{a}\right)^3-\frac{4v^2}{2}=0\). In this paper, we determine the best possible estimate on the real \(\beta\) so that for some analytic \(p\) satisfying \(p(0)=1\) the following subordination-implication holds:
\[
1+\beta\frac{zp'(z)}{p^j(z)}\prec\mathcal{F}(z)\Rightarrow p(z)\prec\varphi_{Ne}(z),\quad j=0,1,2,
\]
where \(F(z)\) is some Carathéodory function with special geometries like right/left-half of Bernoulli's lemniscate, cardioid, lune, eight-shaped, etc. As applications, we establish sufficient conditions for the Ma-Minda family of nephroid starlike functions given by
\[
\mathcal{S}_{Ne}^*:=\bigg\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)}\prec\varphi_{Ne}(z) \bigg\}.
\]Generalization of the Landau and Becker-Pommerenke inequalitieshttps://zbmath.org/1502.300692023-02-24T16:48:17.026759Z"Kudryavtseva, O. S."https://zbmath.org/authors/?q=ai:kudryavtseva.olga-sergeevna"Solodov, A. P."https://zbmath.org/authors/?q=ai:solodov.aleksei-petrovichSummary: A generalization of the Landau and Becker-Pommerenke inequalities, which are used to solve the problem of sharp domains of univalence in subclasses of holomorphic maps, is obtained.Bounded turning in Möbius structureshttps://zbmath.org/1502.300702023-02-24T16:48:17.026759Z"Aseev, V. V."https://zbmath.org/authors/?q=ai:aseev.vladislav-vasilevichSummary: We study a Möbius-invariant generalization, called BTR, of the classical property of bounded turning in a metric space which was introduced by \textit{P. Tukia} and \textit{J. Väisälä} [Ann. Acad. Sci. Fenn., Ser. A I, Math. 5, 97--114 (1980; Zbl 0403.54005)] and suitable for use in Ptolemaic Möbius structures in the sense of Buyalo. In particular, we prove that every continuum with the BTR property, lying on the boundary of a domain in the complex plane, is locally connected.On existence of Becker extensionhttps://zbmath.org/1502.300712023-02-24T16:48:17.026759Z"Gumenyuk, Pavel"https://zbmath.org/authors/?q=ai:gumenyuk.pavelThe author states a connection between a \(q\)-quasiconformal extension of any univalent function and its Becker extension. Let \(S\) be the class of all holomorphic univalent functions \(f\) in the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\) normalized by \(f(0)=0\), \(f'(0)=1\). A Herglotz function \(p:\mathbb D\times[0,\infty)\to\mathbb C\), \(p(0)=1\), is locally integrable on \([0,\infty)\) for every \(z\in\mathbb D\), and for almost every \(t\geq0\), \(p(\cdot,t)\) is holomorphic in \(\mathbb D\) and \(\text{Re}\,p(\cdot,t)\geq0\). For any \(f\in S\) there exists a Herglotz function \(p\) such that \(f(z)=\lim_{t\to\infty}w(z,t)/w'(0,t)\), \(z\in\mathbb D\), where \(w(z,t)\) is the unique solution of the Loewner-Kufarev equation
\[
\frac{dw}{dt}=-wp(w,t),\;\;t\geq0,\;\;w(z,0)=z\in\mathbb D.
\]
A univalent function \(f\) in a domain \(D\subset\mathbb C\) admits a \(k\)-quasiconformal extension to \(\mathbb C\) if there exists a \(k\)-quasiconformal mapping \(F:\mathbb C\to\mathbb C\) such that \(f=F|_D\). Denote by \(S_k\) the class of all \(f\in S\) admitting \(k\)-quasiconformal extensions to \(\mathbb C\). Let \(k\in[0,1)\) and let \((f_t)\) be a Loewner chain generated by the Loewner-Kufarev equation \(\partial f_t(z)/\partial t=zf'(z,t)p(z,t)\), \(z\in\mathbb D\), \(t\geq0\), whose Herglotz function \(p\) satisfies
\[
p(\mathbb D,t)\subset U(k):=\left\{w\in\mathbb C:\left|\frac{w-1}{w+1}\right|\leq k\right\}\;\;\text{for almost every}\;t\geq0.
\]
The extension \(F\) of \(f=f_0\) is called a Becker extension of \(f\in S\). The main result of the paper is given in the following theorem.
Theorem 1. For every \(q\in(0,1/3)\) there exists \(k_0\in(0,1)\) depending only on \(q\) such that every \(f\in S_q\) admits a \(k_0\)-quasiconformal Becker extension.
Reviewer: Dmitri V. Prokhorov (Saratov)Quasiconformal harmonic mappings between the unit ball and a spatial domain with \(C^{1, \alpha}\) boundaryhttps://zbmath.org/1502.300722023-02-24T16:48:17.026759Z"Gjokaj, Anton"https://zbmath.org/authors/?q=ai:gjokaj.anton"Kalaj, David"https://zbmath.org/authors/?q=ai:kalaj.davidSummary: We prove the following. If \(f\) is a harmonic quasiconformal mapping between the unit ball in \(\mathbb{R}^n\) and a spatial domain with \(C^{1, \alpha}\) boundary, then \(f\) is Lipschitz continuous in \(B\). This generalizes some known results for \(n = 2\) and improves some others in higher dimensional case.Sharp cohomological bound for uniformly quasiregularly elliptic manifoldshttps://zbmath.org/1502.300732023-02-24T16:48:17.026759Z"Kangasniemi, Ilmari"https://zbmath.org/authors/?q=ai:kangasniemi.ilmariSummary: We show that if a compact, connected, and oriented \(n\)-manifold \(M\) without boundary admits a non-constant non-injective uniformly quasiregular self-map, then the dimension of the real singular cohomology ring \(H^*(M;\mathbb{R})\) of \(M\) is bounded from above by \(2^n\). This is a positive answer to a dynamical counterpart of the Bonk-Heinonen conjecture on the cohomology bound for quasiregularly elliptic manifolds. The proof is based on an intermediary result that, if \(M\) is not a rational homology sphere, then each such uniformly quasiregular self-map on \(M\) has a Julia set of positive Lebesgue measure.Fibers of monotone maps of finite distortionhttps://zbmath.org/1502.300742023-02-24T16:48:17.026759Z"Kangasniemi, Ilmari"https://zbmath.org/authors/?q=ai:kangasniemi.ilmari"Onninen, Jani"https://zbmath.org/authors/?q=ai:onninen.janiSummary: We study topologically monotone surjective \(W^{1,n}\)-maps of finite distortion \(f :\Omega \rightarrow \Omega'\), where \(\Omega\), \(\Omega'\) are domains in \(\mathbb{R}^n\), \(n \geq 2\). If the outer distortion function \(K_f \in L_{\mathrm{loc}}^p (\Omega)\) with \(p \geq n-1\), then any such map \(f\) is known to be homeomorphic, and hence the fibers \(f^{-1}\{ y\}\) are singletons. We show that as the exponent of integrability \(p\) of the distortion function \(K_f\) increases in the range \(1/(n-1) \leq p < n-1\), then for increasingly many \(k \in \{ 0,\ldots, n\}\) depending on \(p\), the \(k\):th rational homology group \(H_k (f^{-1}\{ y\}; \mathbb{Q})\) of any reasonably tame fiber \(f^{-1}\{ y\}\) of \(f\) is equal to that of a point. In particular, if \(p \geq (n-2)/2\) then this is true for all \(k \in \{ 0,\ldots, n\}\). We also formulate a Sobolev realization of a topological example by Bing of a monotone \(f :\mathbb{R}^3 \rightarrow\mathbb{R}^3\) with homologically non-trivial fibers. This example has \(K_f \in L^{1/2-\varepsilon}_{\mathrm{loc}}(\mathbb{R}^3)\) for all \(\varepsilon > 0\), which shows that our result is sharp in the case \(n = 3\).On the heterogeneous distortion inequalityhttps://zbmath.org/1502.300752023-02-24T16:48:17.026759Z"Kangasniemi, Ilmari"https://zbmath.org/authors/?q=ai:kangasniemi.ilmari"Onninen, Jani"https://zbmath.org/authors/?q=ai:onninen.janiSummary: We study Sobolev mappings \(f \in W_{\text{loc}}^{1,n} (\mathbb{R}^n, \mathbb{R}^n)\), \(n \ge 2\), that satisfy the heterogeneous distortion inequality
\[
\left| Df(x) \right|^n \le K J_f(x) + \sigma^n(x) \left| f(x) \right|^n
\] for almost every \(x \in \mathbb{R}^n\). Here \(K \in [1, \infty)\) is a constant and \(\sigma \ge 0\) is a function in \(L^n_\text{loc}(\mathbb{R}^n)\). Although we recover the class of \(K\)-quasiregular mappings when \(\sigma \equiv 0\), the theory of arbitrary solutions is significantly more complicated, partly due to the unavailability of a robust degree theory for non-quasiregular solutions. Nonetheless, we obtain a Liouville-type theorem and the sharp Hölder continuity estimate for all solutions, provided that \(\sigma \in L^{n-\varepsilon}(\mathbb{R}^n) \cap L^{n+\varepsilon }(\mathbb{R}^n)\) for some \(\varepsilon >0\). This gives an affirmative answer to a question of \textit{K. Astala} et al. [Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton, NJ: Princeton University Press (2009; Zbl 1182.30001)].On the condensers with variable plates, potential levels and domain of definitionhttps://zbmath.org/1502.300762023-02-24T16:48:17.026759Z"Dubinin, V. N."https://zbmath.org/authors/?q=ai:dubinin.vladimir-n"Kim, V. Yu."https://zbmath.org/authors/?q=ai:kim.v-yu.1Summary: The asymptotic formula is obtained for the capacity of a generalized condenser when parts of its plates contract to prescribed points. We consider condenser with variable potential levels and a set of definition that tends to a predetermined domain.Schwarz lemma for harmonic mappings into a geodesic line in a Riemann surfaceshttps://zbmath.org/1502.300772023-02-24T16:48:17.026759Z"Kalaj, David"https://zbmath.org/authors/?q=ai:kalaj.davidSummary: We prove a Schwarz type lemma for harmonic mappings between the unit disk and a geodesic line in a Riemann surface. It presents a certain extension of the Schwarz lemma for the real harmonic mappings with respect to the Euclidean metric defined on the unit disk.On the continuity of half-plane capacity with respect to Carathéodory convergencehttps://zbmath.org/1502.300782023-02-24T16:48:17.026759Z"Murayama, Takuya"https://zbmath.org/authors/?q=ai:murayama.takuyaSummary: We study the continuity of half-plane capacity as a function of boundary hulls with respect to the Carathéodory convergence. In particular, our interest lies in the case that hulls are unbounded. Under the assumption that every hull is contained in a fixed hull with finite imaginary part and finite half-plane capacity, we show that the half-plane capacity is indeed continuous. We also discuss the extension of this result to the case that the underlying domain is finitely connected.
For the entire collection see [Zbl 1493.11005].On indicator and type of an entire function with roots lying on a rayhttps://zbmath.org/1502.300792023-02-24T16:48:17.026759Z"Braichev, G. G."https://zbmath.org/authors/?q=ai:braichev.georgii-genrikhovich"Sherstyukov, V. B."https://zbmath.org/authors/?q=ai:sherstyukov.vladimir-borisovichSummary: A well-known extremal problem is considered: find the exact lower bound for all possible types of entire functions of the order \(\rho\in(1,+\infty)\setminus\mathbb{N}\) with roots on a ray, under the assumption that a lower density of roots is zero, and the upper one takes the given positive value. An approach to this problem is proposed. It based on the study indicator behavior of such entire functions. For the extremal value for any non-integer \(\rho>1\), the best known two-sided estimate is proved. Theoretical statements are supported by the results of numerical calculations.Some remarks on primeness and dynamics of some classes of entire functionshttps://zbmath.org/1502.300802023-02-24T16:48:17.026759Z"Charak, Kuldeep Singh"https://zbmath.org/authors/?q=ai:charak.kuldeep-singh"Kumar, Manish"https://zbmath.org/authors/?q=ai:kumar.manish.1|kumar.manish.4|kumar.manish.2|kumar.manish|kumar.manish.3"Singh, Anil"https://zbmath.org/authors/?q=ai:singh.anilSummary: In this paper we investigate the primeness of a class of entire functions and discuss the dynamics of a periodic member \(f\) of this class with respect to a transcendental entire function \(g\) that permutes with \(f\). In particular we show that the Julia sets of \(f\) and \(g\) are identical.Generalization of proximate order and applicationshttps://zbmath.org/1502.300812023-02-24T16:48:17.026759Z"Chyzhykov, Igor"https://zbmath.org/authors/?q=ai:chyzhykov.igor"Filevych, Petro"https://zbmath.org/authors/?q=ai:filevych.petro-v"Rättyä, Jouni"https://zbmath.org/authors/?q=ai:rattya.jouniSummary: We introduce a concept of a quasi proximate order which is a generalization of a proximate order and allows us to study efficiently analytic functions whose order and lower order of growth are different. We prove an existence theorem for a quasi proximate order, i.e. a counterpart of Valiron's theorem for a proximate order. As applications, we generalize and complement some results of M. Cartwright and C. N. Linden on asymptotic behavior of analytic functions in the unit disc.Exponential polynomials as solutions of nonlinear differential-difference equationshttps://zbmath.org/1502.300822023-02-24T16:48:17.026759Z"Gao, L. K."https://zbmath.org/authors/?q=ai:gao.linkui"Liu, K."https://zbmath.org/authors/?q=ai:liu.kai.1"Liu, X. L."https://zbmath.org/authors/?q=ai:liu.xinlingSummary: Exponential polynomials, an important subclass of finite order entire functions, as solutions of differential or difference or differential-difference equations are considered in [\textit{J. Heittokangas} et al., Ann. Acad. Sci. Fenn., Math. 40, No. 2, 985--1003 (2015; Zbl 1337.34089); the second author, Mediterr. J. Math. 13, No. 5, 3015--3027 (2016 Zbl 1355.34121); \textit{Z. T. Wen} et al., Acta Math. Sin., Engl. Ser. 28, No. 7, 1295--1306 (2012; Zbl 1260.39027); \textit{Z.-T. Wen} et al., J. Differ. Equations 264, No. 1, 98--114 (2018; Zbl 1387.34122)]. The critical domains of zeros and the quotients of exponential polynomials are considered in [\textit{J. Heittokangas} et al., Isr. J. Math. 227, No. 1, 397--421 (2018; 1412.30113)]. In this paper, we proceed to consider the exponential polynomials as solutions of some general complex differential-difference equations and extend existence results.A Turán-type inequality for entire functions of exponential typehttps://zbmath.org/1502.300832023-02-24T16:48:17.026759Z"Shah, Wali Mohammad"https://zbmath.org/authors/?q=ai:shah.wali-mohammad"Singh, Sooraj"https://zbmath.org/authors/?q=ai:singh.soorajSummary: Let \(f(z)\) be an entire function of exponential type \(\tau\) such that \(\|f\| = 1\). Also suppose, in addition, that \(f(z)\neq 0\) for \(\mathfrak{I}z > 0\) and that \(h_f(\frac{\pi}{2}) = 0\). Then, it was proved by \textit{R. B. Gardner} and \textit{N. K. Govil} [Proc. Am. Math. Soc. 123, No. 9, 2757--2761 (1995; ZBl 0841.30024)] that for
\(y= \mathfrak{I} z\le 0\)
\[
\|D_{\zeta}[f]\| \leq \frac{\tau}{2} (|\zeta|+1),
\] where \(D_{\zeta}[f]\) is referred to as polar derivative of entire function \(f(z)\) with respect to \(\zeta \). In this paper, we prove an inequality in the opposite direction and thereby obtain some known inequalities concerning polynomials and entire functions of exponential type.Ratios of entire functions and generalized Stieltjes functionshttps://zbmath.org/1502.300842023-02-24T16:48:17.026759Z"Askitis, Dimitris"https://zbmath.org/authors/?q=ai:askitis.dimitris"Pedersen, Henrik L."https://zbmath.org/authors/?q=ai:pedersen.henrik-laurbergSummary: Monotonicity properties of the ratio
\[
\log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)},
\]
where \(f\) is an entire function are investigated. Earlier results for Euler's gamma function and other entire functions of genus 1 are generalised to entire functions of genus \(p\) with negative zeros. Derivatives of order comparable to \(p\) of the expression above are related to generalised Stieltjes functions of order \(p+1\). Our results are applied to the Barnes multiple gamma functions. We also show how recent results on the behaviour of Euler's gamma function on vertical lines can be sharpened and generalised to functions of higher genus. Finally a connection to the so-called Prouhet-Tarry-Escott problem is described.Entire solutions of one certain type of nonlinear differential-difference equationshttps://zbmath.org/1502.300852023-02-24T16:48:17.026759Z"Chen, Wei"https://zbmath.org/authors/?q=ai:chen.wei.4"Thin, Nguyen Van"https://zbmath.org/authors/?q=ai:thin.nguyen-van"Wang, Qiongyan"https://zbmath.org/authors/?q=ai:wang.qiongyanSummary: We describe the entire solutions for two kinds of nonlinear differential-difference equations of the form \[ f^n (z) + \omega f^{n - 1} (z)f^\prime (z) + q_1 (z)e^{Q_1 (z)} f_{c_1} + q_2 (z)e^{Q_2 (z)} f_{c_2} = u(z)e^{v(z)} ,\quad n \geq 3\] and \[f^n (z) + q(z)e^{Q(z)} f(z + c) = u(z)e^{v(z)} ,\quad n \geq 2,\] where \(q, Q, u, v, q_j (z), Q_j (z)\), for \(j = 1, 2\), are polynomials such that \(Q(z)\) and at least one of \(Q_j (z)\) are not constants, \(q(z)\) and \(q_j (z)\) are not identically zero, and \(\omega, c_1, c_2, c\) are constants. Our results improve and generalize some previous results.Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in \(\mathbb{C}^2\)https://zbmath.org/1502.300862023-02-24T16:48:17.026759Z"Haldar, Goutam"https://zbmath.org/authors/?q=ai:haldar.goutam"Ahamed, Molla Basir"https://zbmath.org/authors/?q=ai:ahamed.molla-basirSummary: An equation is called a complex partial differential-difference equation, if this equation includes partial derivatives, shifts, or differences of complex-valued functions \(f\), which can be called \textit{PDDE} for short and a functional equation of the form \(f^n+g^n = 1\), where \(n\) is an integer, is called the Fermat-type equation. In this paper, we study the existence and the precise form of finite order transcendental entire solutions of several Fermat-type \textit{PDDEs} and quadratic trinomial functional equations in \(\mathbb{C}^2\). The main results of the paper generalize several existing results in this direction. In addition, we exhibited by several examples that our results are precise to some extent.Uniqueness of some differential-difference polynomials of entire functions sharing small functionhttps://zbmath.org/1502.300872023-02-24T16:48:17.026759Z"Naveenkumar, S. H."https://zbmath.org/authors/?q=ai:naveenkumar.s-h(no abstract)\(\phi\)-relative order and \(\phi\)-relative type of meromorphic functions in \(\mathbb{C}\)https://zbmath.org/1502.300882023-02-24T16:48:17.026759Z"Adud, Md."https://zbmath.org/authors/?q=ai:adud.md"Tamang, Samten"https://zbmath.org/authors/?q=ai:tamang.samtenSummary: Let \(\phi\) be a strictly increasing unbounded function on \(\left[1,\infty \right)\). In the paper, we have defined \(\phi\)-relative order and \(\phi\)-relative type of meromorphic functions with respect to an entire function. We have establised some algebraic properties of them and their applications which can be used to study arbitrary growth of analytic or meromorphic functions.Unique range sets without Fujimoto's hypothesishttps://zbmath.org/1502.300892023-02-24T16:48:17.026759Z"Chakraborty, Bikash"https://zbmath.org/authors/?q=ai:chakraborty.bikashSummary: This paper studies the uniqueness of two non-constant meromorphic functions when they share a finite set. Moreover, we will give an existence of unique range sets for meromorphic functions that are the zero sets of some polynomials that do not necessarily satisfy the Fujimoto hypothesis [\textit{H. Fujimoto}, Am. J. Math. 122, No. 6, 1175--1203 (2000; 0983.30013)].Meromorphic solutions of some non-linear \(q\)-shift difference equationshttps://zbmath.org/1502.300902023-02-24T16:48:17.026759Z"Dyavanal, Renukadevi S."https://zbmath.org/authors/?q=ai:dyavanal.renukadevi-sangappa"Muttagi, Jyoti B."https://zbmath.org/authors/?q=ai:muttagi.jyoti-b"N., Shilpa"https://zbmath.org/authors/?q=ai:n.shilpa(no abstract)Unicity of meromorphic functions concerning differences and small functionshttps://zbmath.org/1502.300912023-02-24T16:48:17.026759Z"He, Zhiying"https://zbmath.org/authors/?q=ai:he.zhiying"Xiao, Jianbin"https://zbmath.org/authors/?q=ai:xiao.jianbin"Fang, Mingliang"https://zbmath.org/authors/?q=ai:fang.mingliang(no abstract)On uniqueness of meromorphic functions and their derivativeshttps://zbmath.org/1502.300922023-02-24T16:48:17.026759Z"Meng, Chao"https://zbmath.org/authors/?q=ai:meng.chao"Li, Xu"https://zbmath.org/authors/?q=ai:li.xu(no abstract)Entire solutions of logistic type delay differential equationshttps://zbmath.org/1502.300932023-02-24T16:48:17.026759Z"Nagaswara, P."https://zbmath.org/authors/?q=ai:nagaswara.p"Rajeshwari, S."https://zbmath.org/authors/?q=ai:rajeshwari.s"Chand, M."https://zbmath.org/authors/?q=ai:chand.mehar|chand.maheshSummary: The main purpose of this paper is to study the existence of admissible solutions of logistic delay differential equation of the form
\[
w^\prime(z)= w(z)[R(z,w(z)) + \sum_{j=1}^k b_j (z) w(z - c_j)]
\] by making use of Nevanlinna theory in complex analysis.Study of Brück conjecture and uniqueness of rational function and differential polynomial of a meromorphic functionhttps://zbmath.org/1502.300942023-02-24T16:48:17.026759Z"Pramanik, Dilip Chandra"https://zbmath.org/authors/?q=ai:pramanik.dilip-chandra"Roy, Jayanta"https://zbmath.org/authors/?q=ai:roy.jayantaSummary: Let \(f\) be a non-constant meromorphic function in the open complex plane \(\mathbb{C}\). In this paper we prove under certain essential conditions that \(R(f)\) and \(P[f]\), rational function and differential polynomial of \(f\) respectively, share a small function of \(f\) and obtain a conclusion related to Brück conjecture. We give some examples in support to our result.A new subclass of meromorphic functions with positive coefficients defined by polylogarithm functionhttps://zbmath.org/1502.300952023-02-24T16:48:17.026759Z"Venkateswarlu, Bollineni"https://zbmath.org/authors/?q=ai:venkateswarlu.bollineniSummary: In this paper, we introduce and study a new subclass of meromorphic univalent functions defined by Bessel function. We obtain coefficient inequalities, extreme points, radius of starlikeness and convexity. Finally we obtain partial sums and neighborhood properties for the class \(\sigma_p^\ast(\eta , k, \lambda, m , \upsilon , c)\).Pole recovery from noisy data on imaginary axishttps://zbmath.org/1502.300962023-02-24T16:48:17.026759Z"Ying, Lexing"https://zbmath.org/authors/?q=ai:ying.lexingSummary: This note proposes an algorithm for identifying the poles and residues of a meromorphic function from its noisy values on the imaginary axis. The algorithm uses Möbius transform and Prony's method in the frequency domain. Numerical results are provided to demonstrate the performance of the algorithm.Uniqueness of \(P(f)\) and \([P(f)]^{(k)}\) concerning weakly weighted sharinghttps://zbmath.org/1502.300972023-02-24T16:48:17.026759Z"Ahamed, Molla Basir"https://zbmath.org/authors/?q=ai:ahamed.molla-basirSummary: In this paper, with the help of the idea of weakly weighted sharing introduced by \textit{S. Lin} and \textit{W. Lin} [Kodai Math. J. 29, No. 2, 269--280 (2006; Zbl 1105.30017)], we study the uniqueness of polynomial expression \(P(f)\) and its derivatives \([P(f)]^{(k)}\) of meromorphic functions \(f\) sharing a small function. The main results in the paper significantly improved the result of \textit{L. Liu} and \textit{Y. Gu} [Kodai Math. J. 27, No. 3, 272--279 (2004; Zbl 1115.30034)]. This research work finds certain condition under which the polynomial \(P(f)\) is reduced to a non-zero monomial and consequently, the class of the function \(f\) is characterized. Examples have been exhibited to show that some conditions in the main results simply can not be removed and also some inequalities are sharp.Meromorphic functions sharing a set with their differential monomialshttps://zbmath.org/1502.300982023-02-24T16:48:17.026759Z"Ahamed, Molla Basir"https://zbmath.org/authors/?q=ai:ahamed.molla-basirSummary: Let \(S\) be a subset of \(\mathbb{C}\) and for a non-constant meromorphic function \(f\), we define \(E_f(S):=\bigcup_{a\in S}\{z: f(z)-a=0\}\), where each zero is counted according to its multiplicity. If \(E_f(S)=E_g(S)\), then we say that \(f\) and \(g\) share the set \(S\) CM. The main aim of this paper is to discuss the uniqueness of meromorphic functions sharing the set \(\mathcal{S}=\{w:aw^n+bw^{2m}+cw^m+1=0\}\) with their differential monomials, where \(n>2m\) and \(a, b, c\in \mathbb{C}\). Our key findings in this paper is the precise form of the solutions of certain differential equations obtained in the main result. A number of examples have been exhibited to validate certain claims of the main results. As a consequence, we prove a corollary of the main results which improved the corresponding results of \textit{M. Fang} and \textit{L. Zalcman} [J. Math. Anal. Appl. 280, No. 2, 273--283 (2003; Zbl 1030.30025)], and \textit{J. Chang} et al. [Arch. Math. 89, No. 6, 561--569 (2007; 1148.30012)] in some sense.Further investigations on two shared set problems under deficient valueshttps://zbmath.org/1502.300992023-02-24T16:48:17.026759Z"Banerjee, Abhijit"https://zbmath.org/authors/?q=ai:banerjee.abhijit"Kundu, Arpita"https://zbmath.org/authors/?q=ai:kundu.arpitaSummary: With the aid of different deficient values, we have established some uniqueness theorems for meromorphic functions sharing two sets. Our results have improved a number of earlier results such as [the first author, Publ. Inst. Math., Nouv. Sér. 92(106), 177--187 (2012; Zbl 1289.30176)] and [\textit{I. Lahiri}, Arch. Math., Brno 38, No. 2, 119--128 (2002; Zbl 1087.30028)] in some sense. We have also provided two examples to show the sharpness of our result.Lower order for meromorphic solutions to linear delay-differential equationshttps://zbmath.org/1502.301002023-02-24T16:48:17.026759Z"Bellaama, Rachid"https://zbmath.org/authors/?q=ai:bellaama.rachid"Belaïdi, Benharrat"https://zbmath.org/authors/?q=ai:belaidi.benharratSummary: In this article, we study the order of growth for solutions of the non-homogeneous linear delay-differential equation \[ \sum_{i=0}^n \, \sum_{j=0}^m A_{ij} f^{(j)} (z+c_i)=F(z), \] where \(A_{ij}(z) \,\, (i=0, \ldots, n; j=0, \ldots, m)\), \(F(z)\) are entire or meromorphic functions and \(c_i (0, 1, \ldots, n)\) are non-zero distinct complex numbers. Under the condition that there exists one coefficient having the maximal lower order, or having the maximal lower type, strictly greater than the order, or the type, of the other coefficients, we obtain estimates of the lower bound of the order of meromorphic solutions of the above equation.On the growth measures of entire functions focusing their generalized relative order \((\alpha, \beta)\) and generalized relative type \((\alpha, \beta)\)https://zbmath.org/1502.301012023-02-24T16:48:17.026759Z"Biswas, Tanmay"https://zbmath.org/authors/?q=ai:biswas.tanmay"Biswas, Chinmay"https://zbmath.org/authors/?q=ai:biswas.chinmay(no abstract)A note on the integral representations of generalized relative ORDER \((\alpha, \beta)\) and generalized relative TYPE \((\alpha, \beta)\) of entire and meromorphic functions with respect to an entire functionhttps://zbmath.org/1502.301022023-02-24T16:48:17.026759Z"Biswas, Tanmay"https://zbmath.org/authors/?q=ai:biswas.tanmay"Biswas, Chinmay"https://zbmath.org/authors/?q=ai:biswas.chinmaySummary: In this paper we wish to establish the integral representations of generalized relative order \((\alpha,\beta)\) and generalized relative type \((\alpha,\beta)\) of entire and meromorphic functions where \(\alpha\) and \(\beta\) are continuous non-negative functions defined on \((-\infty,+\infty)\). We also investigate their equivalence relation under some certain condition.Uniqueness of meromorphic functions concerning their derivatives and shifts with partially shared valueshttps://zbmath.org/1502.301032023-02-24T16:48:17.026759Z"Chen, W.-J."https://zbmath.org/authors/?q=ai:chen.weiji|chen.wenjing|chen.wei-james|chen.wujun|chen.wenju|chen.wenji|chen.weijin|chen.weijun|chen.wen-jen|chen.wan-jin|chen.wenjuan|chen.wanji|chen.wang-ji|chen.weijiong|chen.weijian|chen.wengju|chen.wenjie|chen.weijiao|chen.wen-jyh|chen.wen-jiang|chen.weijie|chen.wei-ju|chen.wenjian|chen.weijia|chen.wen-jun|chen.wen-jinn|chen.weijiang"Huang, Z.-G."https://zbmath.org/authors/?q=ai:huang.zhonggan|huang.zhengui|huang.zhigou|huang.zhengge|huang.ze-gui|huang.zhenggang|huang.zhigong|huang.zi-gang|huang.zhengguo|huang.zhiguan|huang.zhengan|huang.zhi-guo|huang.zengguang|huang.zhenguang|huang.zhigao|huang.zhengao|huang.zhigangSummary: The uniqueness problems of the \(j\)th derivative of a meromorphic function \(f(z)\) and the \(k\)th derivative of its shift \(f(z+c)\) are investigated in this paper, where \(j,k\) are integers with \(0\leqslant j<k\). We show that when \(f^{(j)}(z)\) and \(f^{(k)}(z+c)\) share one IM value and two partially shared values CM, the uniqueness result remains valid under some additional hypotheses. With one CM value and two partially shared values CM, a uniqueness theorem about the \(j\)th derivative of \(f(z)\) and the \(k\)th derivative of its shift \(f(z+c)\) is also proved.Nevanlinna theory via holomorphic formshttps://zbmath.org/1502.301042023-02-24T16:48:17.026759Z"Dong, Xianjing"https://zbmath.org/authors/?q=ai:dong.xianjing"Yang, Shuangshuang"https://zbmath.org/authors/?q=ai:yang.shuangshuangSummary: This paper redevelops Nevanlinna theory for meromorphic functions on \(\mathbb{C}\) in the viewpoint of holomorphic forms. According to our observation, Nevanlinna's functions can be formulated by a holomorphic form. Applying this thought to Riemann surfaces, one then extends the definition of Nevanlinna's functions using a holomorphic form \(\mathscr{S}\). With the new settings, an analogue of Nevanlinna theory for the \emph{\( \mathscr{S}\)-exhausted Riemann surfaces} is obtained, which is viewed as a generalization of the classical Nevanlinna theory for \(\mathbb{C}\) and \(\mathbb{D}\).Uniqueness of higher order \(c\)-shift difference polynomials of meromorphic functions with weighted sharinghttps://zbmath.org/1502.301052023-02-24T16:48:17.026759Z"Dyavanal, Renukadevi S."https://zbmath.org/authors/?q=ai:dyavanal.renukadevi-sangappa"Muttagi, Jyoti B."https://zbmath.org/authors/?q=ai:muttagi.jyoti-bSummary: In this paper, we investigate the uniqueness of higher order \(c\)-shift difference operator of meromorphic functions sharing the value 1 with weight \(l\). Our results extend and generalize the results of \textit{R. S. Dyavanal} and \textit{M. M. Mathai} [Ukr. Math. J. 71, No. 7, 1032--1042 (2019); translation from Ukr. Mat. Zh. 71, No. 7, 906--914 (2019; Zbl 1435.30096)].Limit behavior of Weyl coefficientshttps://zbmath.org/1502.301062023-02-24T16:48:17.026759Z"Pruckner, R."https://zbmath.org/authors/?q=ai:pruckner.raphael"Woracek, H."https://zbmath.org/authors/?q=ai:woracek.haraldSummary: The sets of radial or nontangential limit points towards \(i\infty\) of a Nevanlinna function \(q\) are studied. Given a nonempty, closed, and connected subset \({\mathcal{L}}\) of \(\overline{{\mathbb{C}}_+} \), a Hamiltonian \(H\) is constructed explicitly such that the radial and outer angular cluster sets towards \(i\infty\) of the Weyl coefficient \(q_H\) are both equal to \({\mathcal{L}} \). The method is based on a study of the continuous group action of rescaling operators on the set of all Hamiltonians.The value distribution of meromorphic functions with relative \((k, n)\) Valiron defect on annulihttps://zbmath.org/1502.301072023-02-24T16:48:17.026759Z"Rathod, A."https://zbmath.org/authors/?q=ai:rathod.ashok-meghappa|rathod.a-k|rathod.abhishekSummary: In the paper, we study and compare relative \((k,n)\) Valiron defect with the relative Nevanlinna defect for meromorphic function where \(k\) and \(n\) are both non negative integers on annuli. The results we proved are as follows
1. Let \(f(z)\) be a transcendental or admissible meromorphic function of finite order in \(\mathbb{A}(R_0)\), where \(1<R_0\leq +\infty\) and \(\sum\nolimits_{a\not=\infty}\delta_0(a,f)+\delta_0(\infty,f)=2\). Then
\[\lim\limits_{R\rightarrow\infty}^{}\frac{T_0(R,f^{(k)})}{T_0(R,f)}=(1+k)-k\delta_0(\infty,f).\]
2. Let \(f(z)\) be a transcendental or admissible meromorphic function of finite order in \(\mathbb{A}(R_0)\), where \(1<R_0\leq +\infty\) such that \(m_0(r,f)=S(r,f)\). If \(a\), \(b\) and \(c\) are three distinct complex numbers, then for any two positive integer \(k\) and \(n\)
\[3_R\delta_{0(n)}^{(0)}(a,f)+2_R\delta_{0(n)}^{(0)}(b,f)+3_R\delta_{0(n)}^{(0)}(c,f)+5_R\Delta_{0(n)}^{(k)}(\infty ,f)\leq 5_R\Delta_{0(n)}^{(0)}(\infty,f)+5_R\Delta_{0(n)}^{(k)}(0,f).\]
3. Let \(f(z)\) be a transcendental or admissible meromorphic function of finite order in \(\mathbb{A}(R_0)\), where \(1<R_0\leq +\infty\) such that \(m_0(r,f)=S(r,f)\). If \(a\), \(b\) and \(c\) are three distinct complex numbers, then for any two positive integer \(k\) and \(n\)
\[{}_R\delta_{0(n)}^{(0)}(0,f)+_R\Delta_{0(n)}^{(k)}(\infty,f)+_R\delta_{0(n)}^{(0)}(c,f)\leq_R\Delta_{0(n)}^{(0)}(\infty,f)+2_R\Delta_{0(n)}^{(k)}(0,f).\]
4. Let \(f(z)\) be a transcendental or admissible meromorphic function of finite order in \(\mathbb{A}(R_0)\), where \(1<R_0\leq +\infty\) such that \(m_0(r,f)=S(r,f)\). If \(a\) and \(d\) are two distinct complex numbers, then for any two positive integer \(k\) and \(p\) with \(0\leq k\leq p\)
\[{}_R\delta_{0(n)}^{(0)}(d,f)+_R\Delta_{0(n)}^{(p)}(\infty,f)+_R\delta_{0(n)}^{(k)}(a,f)\leq_R\Delta_{0(n)}^{(k)}(\infty,f)+_R\Delta_{0(n)}^{(p)}(0,f)+_R\Delta_{0(n)}^{(k)}(0,f),\]
where \(n\) is any positive integer.
5. Let \(f(z)\) be a transcendental or admissible meromorphic function of finite order in \(\mathbb{A}(R_0)\), where \(1<R_0\leq +\infty \). Then for any two positive integers \(k\) and \(n\),
\[{}_R\Delta_{0(n)}^{(0)}(\infty,f)+_R\Delta_{0(n)}^{(k)}(0,f) \geq_R\delta_{0(n)}^{(0)}(0,f)+_R\delta_{0(n)}^{(0)}(a,f)+_R\Delta_{0(n)}^{(k)}(\infty,f),\]
where \(a\) is any non zero complex number.Uniqueness of a polynomial and differential polynomial sharing a small functionhttps://zbmath.org/1502.301082023-02-24T16:48:17.026759Z"Waghamore, Harina P."https://zbmath.org/authors/?q=ai:waghamore.harina-pandit"Maligi, Ramya"https://zbmath.org/authors/?q=ai:maligi.ramya(no abstract)Meromorphic solutions of one certain type of non-linear complex differential equationhttps://zbmath.org/1502.301092023-02-24T16:48:17.026759Z"Zhang, Jie"https://zbmath.org/authors/?q=ai:zhang.jie.12"Liao, Liangwen"https://zbmath.org/authors/?q=ai:liao.liangwenSummary: In this paper, we are mainly concerned with one certain type of non-linear complex differential equation
\[
f^n(z)+ P_d(z,f)=p_1 e^{\alpha_1 z} + p_2 e^{\alpha_2 z},
\] where \(p_1,p_2,\alpha_1,\alpha_2\) are non-zero constants and \(P_d(z,f)\) is a differential polynomial in \(f\) of degree \(d\) at most \(n -1\). For this kind of equation, if it admits a meromorphic solution such that \(N(r,f) = S(r,f)\), then firstly we give a positive answer to \textit{P. Li}'s question [J. Math. Anal. Appl. 375, No. 1, 310--319 (2011; ZBl 1206.30046)] with a weaker restriction on \(\alpha_1\) and \(\alpha_2\) than Li's when \(\frac{\alpha_2}{\alpha_1}>0\); secondly, we show \(\alpha_1+\alpha_2=0\) when \(\frac{\alpha_2}{\alpha_1}<0\).On the lambda function and a quantification of Torhorst theoremhttps://zbmath.org/1502.301102023-02-24T16:48:17.026759Z"Feng, Li"https://zbmath.org/authors/?q=ai:feng.li"Luo, Jun"https://zbmath.org/authors/?q=ai:luo.jun"Yao, Xiao-Ting"https://zbmath.org/authors/?q=ai:yao.xiaotingSummary: For any compact \(K \subset \hat{\mathbb{C}}\) we define a map \(\lambda_K : \hat{\mathbb{C}} \to \mathbb{N} \cup \{\infty \}\), called the lambda function of \(K\). The lambda function of a continuum \(K\) has the property that \(\lambda_K(x) = 0\) for all \(x \in \hat{\mathbb{C}}\) if and only if \(K\) is locally connected. We establish several inequalities and a gluing lemma for the lambda functions. These inequalities reflect the relationship between the topology of \(K\) and the complexity that may be involved in the boundary of a component of \(\hat{\mathbb{C}} \setminus K\). One of them, called the lambda inequality, generalizes and quantifies the Torhorst Theorem. We also find three sufficient conditions under each of which the lambda inequality becomes an equality. These results cover Whyburn's Theorem on the partial converse to the Torhorst Theorem. The gluing lemma for the lambda functions is of research interest in the study of point set topology and of polynomial Julia sets as well.On continuous extension of conformal homeomorphisms of infinitely connected planar domainshttps://zbmath.org/1502.301112023-02-24T16:48:17.026759Z"Luo, Jun"https://zbmath.org/authors/?q=ai:luo.jun"Yao, Xiaoting"https://zbmath.org/authors/?q=ai:yao.xiaotingSummary: We consider conformal homeomorphisms \(\varphi\) of generalized Jordan domains \(U\) onto planar domains \(\Omega\) that satisfy both of the next two conditions: (1) at most countably many boundary components of \(\Omega\) are non-degenerate and their diameters have a finite sum; (2) either the degenerate boundary components of \(\Omega\) or those of \(U\) form a set of sigma-finite linear measure. We prove that \(\varphi\) continuously extends to the closure of \(U\) if and only if every boundary component of \(\Omega\) is locally connected. This generalizes the Carathéodory's Continuity Theorem and leads us to a new generalization of the well known Osgood-Taylor-Carathéodory Theorem. There are three issues that are noteworthy. Firstly, none of the above conditions (1) and (2) can be removed. Secondly, our results remain valid for non-cofat domains and do not follow from the extension results, of a similar nature, that are obtained in very recent studies on the conformal rigidity of circle domains. Finally, when \(\varphi\) does extend continuously to the closure of \(U\), the boundary of \(\Omega\) is a Peano compactum. Therefore, we also show that the following properties are equivalent for any planar domain \(\Omega \):
\begin{itemize}
\item[(1)] The boundary of \(\Omega \) is a Peano compactum.
\item[(2)] \(\Omega \) has Property S.
\item[(3)] Every point on the boundary of \(\Omega \) is locally accessible.
\item[(4)] Every point on the boundary of \(\Omega \) is locally sequentially accessible.
\item[(5)] \(\Omega \) is finitely connected at the boundary.
\item[(6)] The completion of \(\Omega \) under the Mazurkiewicz distance is compact.
\end{itemize}Complete interpolating sequences for the Gaussian shift-invariant spacehttps://zbmath.org/1502.301122023-02-24T16:48:17.026759Z"Baranov, Anton"https://zbmath.org/authors/?q=ai:baranov.anton-d"Belov, Yurii"https://zbmath.org/authors/?q=ai:belov.yurii-s"Gröchenig, Karlheinz"https://zbmath.org/authors/?q=ai:grochenig.karlheinzSummary: We give a full description of complete interpolating sequences for the shift-invariant space generated by the Gaussian. As a consequence, we rederive the known density conditions for sampling and interpolation.Marcinkiewicz-Zygmund inequalities for polynomials in Fock spacehttps://zbmath.org/1502.301132023-02-24T16:48:17.026759Z"Gröchenig, Karlheinz"https://zbmath.org/authors/?q=ai:grochenig.karlheinz"Ortega-Cerdà, Joaquim"https://zbmath.org/authors/?q=ai:ortega-cerda.joaquimSummary: We study the relation between Marcinkiewicz-Zygmund families for polynomials in a weighted \(L^2\)-space and sampling theorems for entire functions in the Fock space and the dual relation between uniform interpolating families for polynomials and interpolating sequences. As a consequence we obtain a description of signal subspaces spanned by Hermite functions by means of Gabor frames.Jackson-Stechkin inequality and values of widths of some classes of functions in \(L_2\)https://zbmath.org/1502.301142023-02-24T16:48:17.026759Z"Shabozov, M. Sh."https://zbmath.org/authors/?q=ai:shabozov.mirgand-shabozovich"Palavonov, K. K."https://zbmath.org/authors/?q=ai:palavonov.k-kSummary: The sharp values of extremal characteristic of special form for classes \(L_2^{(r)}\), \((r\in\mathbb{Z}_+)\) containing not only averaged module of continuity but also the averaged with weight \(u(t-u)/t\), \(0\le u\le t\) of given modulus continuity is calculated. The obtained result is the spreading of well-known S. B. Vakarchuk theorem about averaged module of continuity. For the given characteristic of smoothness, is given an application for the solution of one extremal problem and the values of \(n\)-widths for some classes of functions in \(L_2\) is calculated.Pointwise estimation of sign-preserving polynomial approximations on arcs in the complex planehttps://zbmath.org/1502.301152023-02-24T16:48:17.026759Z"Shchehlov, M. V."https://zbmath.org/authors/?q=ai:shchehlov.m-vSummary: In 2014, \textit{V. Andrievskii} [Comput. Methods Funct. Theory 13, No. 3, 493--508 (2013; Zbl 1281.30006)] proved that if a real-valued function \(f \in \operatorname{Lip} \alpha\), \(0 < \alpha < 1\), defined on a given smooth Jordan curve satisfying the Dini condition changes its sign finitely many times, then it can be approximated by a harmonic polynomial that changes its sign on the indicated curve at the same points as \(f\), and the approximation error has the same order as the classical Dzyadyk's error of pointwise approximation. By applying the scheme of the proof proposed by Andrievskii, we generalize his result to the case of an arbitrary modulus of continuity \(\omega(f, t)\) satisfying the inequality \(\gamma \omega ( f, 2t) \geq \omega(f, t)\), where \(\gamma = \mathrm{const} < 1.\)On one equality of integralshttps://zbmath.org/1502.301162023-02-24T16:48:17.026759Z"Dmitriev, A. A."https://zbmath.org/authors/?q=ai:dmitriev.aleksandr-a|dmitriev.a-aSummary: The note proves the equality \[\frac1{2\pi i} \oint_{|z|=r}\ln\left[1{-}f(z)\left(z{+}\frac1z\right)\right]dz= -\frac1{2\pi i}\oint_{|z|=r}\exp\left(\frac{1{-}\sqrt{1-4f^2(z)}}{2zf(z)}\right)dz.\]Diffraction by a right-angled no-contrast penetrable wedge revisited: a double Wiener-Hopf approachhttps://zbmath.org/1502.301172023-02-24T16:48:17.026759Z"Kunz, Valentin D."https://zbmath.org/authors/?q=ai:kunz.valentin-d"Assier, Raphael"https://zbmath.org/authors/?q=ai:assier.raphael-cSummary: In this paper, we revisit Radlow's innovative approach to diffraction by a penetrable wedge by means of a double Wiener-Hopf technique. We provide a constructive way of obtaining his ansatz and give yet another reason for why his ansatz cannot be the true solution to the diffraction problem at hand. The two-complex-variable Wiener-Hopf equation is reduced to a system of two equations, one of which contains Radlow's ansatz plus some correction term consisting of an explicitly known integral operator applied to a yet unknown function, whereas the other equation, the compatibility equation, governs the behavior of this unknown function.The Clairaut-Schwarz theorem for mixed Wirtinger derivativeshttps://zbmath.org/1502.301182023-02-24T16:48:17.026759Z"Mortini, Raymond"https://zbmath.org/authors/?q=ai:mortini.raymond"Rupp, Rudolf"https://zbmath.org/authors/?q=ai:rupp.rudolfSummary: In this note on the foundations of complex analysis, we present for Wirtinger derivatives a short proof of the analogue of the Clairaut-Schwarz theorem. It turns out that, via Fubini's theorem for disks, it is a consequence of the complex version of the Gauss-Green formula relating planar integrals on disks to line integrals on the boundary circle. At the same time, we obtain a version of Pompeiu's formula for partial aerolar derivatives (equivalent to Wirtinger derivatives) in several complex variables.Schwarz boundary value problem on Reuleaux trianglehttps://zbmath.org/1502.301192023-02-24T16:48:17.026759Z"Hao, Yanshuai"https://zbmath.org/authors/?q=ai:hao.yanshuai"Hua, Liu"https://zbmath.org/authors/?q=ai:hua.liuSummary: We discuss the Schwarz problem on Reuleaux triangle. By the technique of parqueting reflection principle, we translate the Schwarz problem into a Riemann boundary value problem on a system of arcs. We first state that the Riemann problem is solvable, while the solutions are represented by a Cauchy type integral. At last, as the main theorem, we prove that those two problems are equivalent to each other. And then we obtain the explicit integral representation of the solution to a Schwarz problem.Determination of the Hall voltage for the case of a Hall plate having piecewise constant Hall anglehttps://zbmath.org/1502.301202023-02-24T16:48:17.026759Z"Homentcovschi, Dorel"https://zbmath.org/authors/?q=ai:homentcovschi.dorel"Murray, Bruce T."https://zbmath.org/authors/?q=ai:murray.bruce-tSummary: A method is developed to determine the Hall voltage for the case of a Hall plate exposed to a piecewise constant magnetic field. The electromagnetic field is discontinuous at the interface between any two subdomains. The conditions satisfied by the electric field on a discontinuity curve (interface) inside a Hall plate with four contacts are obtained as: (a) the continuity of the potential function along the whole plate and (b) a discontinuity relationship for the flux function across the interface in terms of the potential function on the interface. An application of the method is given for the case of four point contacts where the voltage along the interface is computed using a complex variable interface element method. This voltage is used to calculate the Hall voltage of the entire plate structure.Uniformization of Riemann surfaces revisitedhttps://zbmath.org/1502.301212023-02-24T16:48:17.026759Z"Anghel, Cipriana"https://zbmath.org/authors/?q=ai:anghel.cipriana"Stan, Rareş"https://zbmath.org/authors/?q=ai:stan.raresSummary: We give an elementary and self-contained proof of the uniformization theorem for noncompact simply connected Riemann surfaces.Central points of the double heptagon translation surface are not connection pointshttps://zbmath.org/1502.301222023-02-24T16:48:17.026759Z"Boulanger, Julien"https://zbmath.org/authors/?q=ai:boulanger.julienSummary: We consider flow directions on the translation surfaces formed from double \((2n+1)\)-gons and give a sufficient condition in terms of a natural continued fractions algorithm for a direction to be hyperbolic in the sense that it is a fixed direction for some hyperbolic element of the Veech group of the surface. In particular, we give explicit points with coordinates in the trace field of the double heptagon translation surface, that are not so-called connection points. Among these are the central points of the heptagons, giving a negative answer to a question by \textit{P. Hubert} and \textit{T. Schmidt} [Private communication] .Geometric deduction of the solutions to modular equationshttps://zbmath.org/1502.301232023-02-24T16:48:17.026759Z"Alam, Md. Shafiul"https://zbmath.org/authors/?q=ai:alam.md-shafiul"Sugawa, Toshiyuki"https://zbmath.org/authors/?q=ai:sugawa.toshiyukiSummary: In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series and hypergeometric functions. We offer a geometric approach to the proof of those formulae. We emphasize that our proofs are geometric and independent of such identities.On Carleson measures induced by Beltrami coefficients being compatible with Fuchsian groupshttps://zbmath.org/1502.301242023-02-24T16:48:17.026759Z"Huo, Shengjin"https://zbmath.org/authors/?q=ai:huo.shengjinSummary: Let \(\mu\) be a Beltrami coefficient on the unit disk, which is compatible with a finitely generated Fuchsian group \(G\) of the second kind. In this paper we show that if \(\frac{|\mu|^2}{1-|z|^2}\,dx\,dy\) satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain of \(G\), then \(\frac{|\mu|^2}{1-|z|^2}\,dx\,dy\) is a Carleson measure on the unit disk.Projections in moduli spaces of the Kleinian groupshttps://zbmath.org/1502.301252023-02-24T16:48:17.026759Z"Alaqad, Hala"https://zbmath.org/authors/?q=ai:alaqad.hala"Gong, Jianhua"https://zbmath.org/authors/?q=ai:gong.jianhua"Martin, Gaven"https://zbmath.org/authors/?q=ai:martin.gaven-j(no abstract)Effective bilipschitz bounds on drilling and fillinghttps://zbmath.org/1502.301262023-02-24T16:48:17.026759Z"Futer, David"https://zbmath.org/authors/?q=ai:futer.david"Purcell, Jessica S."https://zbmath.org/authors/?q=ai:purcell.jessica-shepherd"Schleimer, Saul"https://zbmath.org/authors/?q=ai:schleimer.saulSummary: We prove explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic \(3\)-manifold \(N\) and the thick part of any of its long Dehn fillings. Given a bilipschitz constant \(J > 1\) and a thickness constant \(\epsilon > 0\), we quantify how long a Dehn filling suffices to guarantee a \(J\)-bilipschitz map on \(\epsilon\)-thick parts. A similar theorem without quantitative control was previously proved by Brock and Bromberg, applying Hodgson and Kerckhoff's theory of cone deformations. We achieve quantitative control by bounding the analytic quantities that control the infinitesimal change in metric during the cone deformation.
Our quantitative results have two immediate applications. First, we relate the Margulis number of \(N\) to the Margulis numbers of its Dehn fillings. In particular, we give a lower bound on the systole of any closed \(3\)-manifold \(M\) whose Margulis number is less than \(0.29\). Combined with Shalen's upper bound on the volume of such a manifold, this gives a procedure to compute the finite list of \(3\)-manifolds whose Margulis numbers are below \(0.29\).
Our second application is to the cosmetic surgery conjecture. Given the systole of a one-cusped hyperbolic manifold \(N\), we produce an explicit upper bound on the length of a slope involved in a cosmetic surgery on \(N\). This reduces the cosmetic surgery conjecture on \(N\) to an explicit finite search.On approximate summation of Poincaré series in the Schottky modelhttps://zbmath.org/1502.301272023-02-24T16:48:17.026759Z"Lyamaev, S. Yu."https://zbmath.org/authors/?q=ai:lyamaev.sergei-yuSummary: For approximate summation of Poincaré theta series in the Schottky model of real hyperelliptic curves, modifications of the Bogatyrev and Schmies algorithms are proposed that reduce the number of summed terms without losing accuracy.Hausdorff dimension and complex hyperbolic Schottky groups: a simplificationhttps://zbmath.org/1502.301282023-02-24T16:48:17.026759Z"Ucan-Puc, Alejandro"https://zbmath.org/authors/?q=ai:ucan-puc.alejandro"Romaña, Sergio"https://zbmath.org/authors/?q=ai:romana-ibarra.sergio-augustoSummary: In the present work, we study the Hausdorff dimension of the limit set of Schottky groups on the boundary of the complex hyperbolic group via the Eigenvalue algorithm as in [\textit{C. T. McMullen}, Am. J. Math. 120, No. 4, 691--721 (1998; Zbl 0953.30026)]. The visual geometry on \(\overline{\mathbb{H}^2_\mathbb{C}}\) allows a direct application of the Eigenvalue algorithm, but at the same time, it hardens the computations. Using the Heisenberg structure on \(\partial{\mathbb{H}}^2_\mathbb{C}\setminus \{\infty\}\) and the Cygan metric, we propose a Markov partition associated with the group that simplifies the application of the Eigenvalue algorithm.Metrics on a surface with bounded total curvaturehttps://zbmath.org/1502.301292023-02-24T16:48:17.026759Z"Li, Yuxiang"https://zbmath.org/authors/?q=ai:li.yuxiang"Sun, Jianxin"https://zbmath.org/authors/?q=ai:sun.jianxin"Tang, Hongyan"https://zbmath.org/authors/?q=ai:tang.hongyanSummary: Let \(g=e^{2u}g_{euc}\) be a conformal metric defined on the unit disk of \(\mathbb{C}\). We give an estimate of \(\|\nabla u\|_{L^{2,\infty}(D_{\frac{1}{2}})}\) when \(\|K(g)\|_{L^1}\) is small and \(\frac{\mu(B_r^g(z),g)}{\pi r^2}<\Lambda\) for any \(r\) and \(z\in D_{\frac{3}{4}}\). Then we use this estimate to study the Gromov-Hausdorff convergence of a conformal metric sequence with bounded \(\|K\|_{L^1}\) and give some applications.Comparison and Möbius quasi-invariance properties of Ibragimov's metrichttps://zbmath.org/1502.301302023-02-24T16:48:17.026759Z"Xu, Xiaoxue"https://zbmath.org/authors/?q=ai:xu.xiaoxue"Wang, Gendi"https://zbmath.org/authors/?q=ai:wang.gendi"Zhang, Xiaohui"https://zbmath.org/authors/?q=ai:zhang.xiaohui.1Summary: For a domain \(D \subsetneq{\mathbb{R}}^n \), Ibragimov's metric is defined as
\[
u_D(x,y) = 2\, \log \frac{|x-y|+\max \{d(x),d(y)\}}{\sqrt{d(x)\,d(y)}}, \quad x,y \in D,
\]
where \(d(x)\) denotes the Euclidean distance from \(x\) to the boundary of \(D\). In this paper, we compare Ibragimov's metric with the classical hyperbolic metric in the unit ball or in the upper half space, and prove sharp comparison inequalities between Ibragimov's metric and some hyperbolic type metrics. We also obtain several sharp distortion inequalities for Ibragimov's metric under some families of Möbius transformations.Counting hyperbolic multigeodesics with respect to the lengths of individual components and asymptotics of Weil-Petersson volumeshttps://zbmath.org/1502.301312023-02-24T16:48:17.026759Z"Arana-Herrera, Francisco"https://zbmath.org/authors/?q=ai:arana-herrera.franciscoSummary: Given a connected, oriented, complete, finite-area hyperbolic surface \(X\) of genus \(g\) with \(n\) punctures, Mirzakhani showed that the number of simple closed multigeodesics on \(X\) of a prescribed topological type and total hyperbolic length \(\leq L\) is asymptotic to a polynomial in \(L\) of degree \(6g-6+2n\) as \(L \to \infty\). We establish asymptotics of the same kind for counts of simple closed multigeodesics that keep track of the hyperbolic length of individual components rather than just the total hyperbolic length, proving a conjecture of Wolpert. The leading terms of these asymptotics are related to limits of Weil-Petersson volumes of expanding subsets of quotients of Teichmüller space. We introduce a framework for computing limits of this kind in terms of purely topological information. We provide two further applications of this framework to counts of square-tiled surfaces and counts of filling closed multigeodesics on hyperbolic surfaces.Plurisubharmonicity and geodesic convexity of energy function on Teichmüller spacehttps://zbmath.org/1502.301322023-02-24T16:48:17.026759Z"Kim, Inkang"https://zbmath.org/authors/?q=ai:kim.inkang"Wan, Xueyuan"https://zbmath.org/authors/?q=ai:wan.xueyuan"Zhang, Genkai"https://zbmath.org/authors/?q=ai:zhang.genkaiThe classical Teichmüller space carries different metrics and energy functions defined by them. In this paper the authors study the convexity of a particular function defined via the Weil-Petersson metric.
For a Riemannian manifold \(M^n\) and a fixed Riemannian surface \(X(z)\) with a hyperbolic metric \(\Phi(z)\) with \(z\) a point in the Teichmüller space of Riemannian surfaces of genus \(g\), it is known that any continuous map \(u_0 : M\rightarrow X(z)\) is homotopic to a unique harmonic map \(u\) unless the image is a point or a closed geodesic. The energy \(E(u)\) of \(u\) is a function of \(z\), so defines a function \(E(z)\) on the Teichmüller space. The main result of the paper is that \(\log E(z)\) is plurisubharmonic. Previously similar property has been shown for the function defining the energy of the harmonic map in a homotopy class of maps \(u_0: X(z)\rightarrow M\) in [\textit{D. Toledo}, Geom. Funct. Anal. 22, No. 4, 1015--1032 (2012; Zbl 1254.32020)].
The main result is proven by careful study of the first and second variation of the energy function. As a corollaries the authors obtain another proofs of some results about the length of geodesics function on Teichmüller space.
Reviewer: Gueo Grantcharov (Miami)On equicontinuity of families of mappings between Riemannian surfaces with respect to prime endshttps://zbmath.org/1502.301332023-02-24T16:48:17.026759Z"Sevost'yanov, E."https://zbmath.org/authors/?q=ai:sevostyanov.evgeny"Dovhopiatyi, O. P."https://zbmath.org/authors/?q=ai:dovhopiatyi.oleksandr-petrovych"Ilkevych, N. S."https://zbmath.org/authors/?q=ai:ilkevych.n-s"Kalenska, V. P."https://zbmath.org/authors/?q=ai:kalenska.v-pSummary: Given a domain of some Riemannian surface, we consider questions related to the possibility of a continuous extension to the boundary of one class of Sobolev mappings. It is proved that such maps have a continuous boundary extension in terms of prime ends, and under some additional restrictions their families are equicontinuous at inner and boundary points of the domain. We have separately considered the cases of homeomorphisms and mappings with branching.Schwarz problem for \(J\)-analytic functions in an ellipsehttps://zbmath.org/1502.301342023-02-24T16:48:17.026759Z"Nikolaev, V. G."https://zbmath.org/authors/?q=ai:nikolaev.vladimir-gennadevichSummary: The Schwarz problem for functions analytic in the sense of Douglis in an ellipse is considered. Necessary and sufficient conditions on the \(\ell \times \ell\) matrix \(J\) and the ellipse \(\Gamma\) are obtained under which the Schwarz problem has a unique solution in Hölder classes. In the case of \(\ell = 2\) and matrices with distinct eigenvalues, the Schwarz problem is reduced to a scalar functional equation. Sufficient conditions on a Jordan basis of \(J\) are obtained under which the Schwarz problem is solvable in an arbitrary ellipse. Matrices \(J\) with eigenvalues lying above and below the real line are considered.On the \(\left\langle\rho_j,~W_j\right\rangle\) generalized completely monotone functionshttps://zbmath.org/1502.301352023-02-24T16:48:17.026759Z"Sahakyan, B. A."https://zbmath.org/authors/?q=ai:sahakyan.b-aSummary: We consider sequences \(\{\rho_j\}_0^\infty\) (\(\rho_0=1, \, \rho_j\geq 1\)), \(\{\alpha_j\}_0^\infty\) (\(\alpha_0=0, \,\alpha_j=1-(1/\rho_j)\)), \(\{W_j(x)\}_0^\infty \in W\), where
\[W=\left\{\left\{W_j(x)\right\}_0^\infty \big{/} W_0(x)\equiv 1,~W_j(x)> 0,~W_j^{\prime}(x)\leq 0,~W_j(x)\in C^\infty[0,a] \right\},\]
\(C^\infty[0,a]\) is the class of functions of infinitely differentiable. For such sequences we introduce systems of operators \(\left\{A_{a,n}^*f\right\}_0^\infty\), \(\left\{\tilde{A}_{a,n}^*f\right\}_0^\infty\) and functions \(\left\{U_{a,n}(x)\right\}_0^\infty\), \(\left\{\Phi_n (x,t)\right\}_0^\infty\). For a certain class of functions a generalization of Taylor-Maclaurin type formulae was obtained. We also introduce the concept of \(\langle\rho_j, W_j\rangle\) generalized completely monotone functions and establish a theorem on their representation.Discrete analytic functions, structured matrices and a new family of moment problemshttps://zbmath.org/1502.301362023-02-24T16:48:17.026759Z"Alpay, Daniel"https://zbmath.org/authors/?q=ai:alpay.daniel"Colombo, Fabrizio"https://zbmath.org/authors/?q=ai:colombo.fabrizio"Diki, Kamal"https://zbmath.org/authors/?q=ai:diki.kamal"Sabadini, Irene"https://zbmath.org/authors/?q=ai:sabadini.irene"Volok, Dan"https://zbmath.org/authors/?q=ai:volok.danSummary: Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel spaces, structured matrices and a generalized moment problem. An important role is played by a Krein space realization result of Dijksma, Langer and de Snoo for functions analytic in a neighborhood of the origin. A key observation is that, using a simple Moebius transform, one can reduce the study of discrete analytic functions in the upper right quadrant to problems of function theory in the open unit disk. As an example, we associate to each finite positive measure on \([0, 2 \pi]\) a discrete analytic function on the right-upper quarter plane with values on the lattice defining a positive definite function. Emphasis is put on the rational case, both when an underlying Carathéodory function is rational and when, in the positive case, the spectral function is rational. The rational case and the general case are linked via the existence of a unitary dilation, possibly in a Krein space.Discrete conformal geometry of polyhedral surfaces and its convergencehttps://zbmath.org/1502.301372023-02-24T16:48:17.026759Z"Luo, Feng"https://zbmath.org/authors/?q=ai:luo.feng"Sun, Jian"https://zbmath.org/authors/?q=ai:sun.jian.4"Wu, Tianqi"https://zbmath.org/authors/?q=ai:wu.tianqiSummary: We prove a result on the convergence of discrete conformal maps to the Riemann mappings for Jordan domains. It is a counterpart of Rodin and Sullivan's theorem on convergence of circle packing mappings to the Riemann mapping in the new setting of discrete conformality. The proof follows the same strategy that Rodin and Sullivan used by establishing a rigidity result for regular hexagonal triangulations of the plane and estimating the quasiconformal constants associated to the discrete conformal maps.The fixed points and cross-ratios of hyperbolic Möbius transformations in bicomplex spacehttps://zbmath.org/1502.301382023-02-24T16:48:17.026759Z"Chen, Litao"https://zbmath.org/authors/?q=ai:chen.litao"Dai, Binlin"https://zbmath.org/authors/?q=ai:dai.binlinSummary: The hyperbolic Möbius transformations, which have been defined and proved to be hyperbolic conformal in Golberg and Luna-Elizarrarás (Math Methods Appl Sci 2020, \url{doi:10.1002/mma.7109}), are generalizations of Möbius transformations in complex space \(\mathbb{C}(i)\) and hyperbolic space \(\mathbb{D}\) to multidimensional hyperbolic space \(\mathbb{D}^n\). In this paper, we study the hyperbolic Möbius transformation in bicomplex space \(\mathbb{BC}\) isomorphic to \(\mathbb{D}^2\) in detail, present a conjugacy classification according to the number of fixed points in \(SL(2, \mathbb{BC})\), and detailedly prove that the cross-ratio is invariant under hyperbolic Möbius transformations. Furthermore, the present paper generalizes the classical results, which have closed relation with fixed points and cross-ratios, to \(\mathbb{BC}\) and may give new energy for the development of hyperbolic Möbius groups.On monogenic reproducing kernel Hilbert spaces of the Paley-Wiener typehttps://zbmath.org/1502.301392023-02-24T16:48:17.026759Z"Dang, Pei"https://zbmath.org/authors/?q=ai:dang.pei"Mai, Weixiong"https://zbmath.org/authors/?q=ai:mai.weixiong"Qian, Tao"https://zbmath.org/authors/?q=ai:qian.taoSummary: In the Clifford algebra setting the present study develops three reproducing kernel Hilbert spaces of the Paley-Wiener type, namely the Paley-Wiener spaces, the Hardy spaces on strips, and the Bergman spaces on strips. In particular, we give spectrum characterizations and representation formulas of the functions in those spaces and estimation of their respective reproducing kernels.Monogenic functions with values in generalized Clifford algebrashttps://zbmath.org/1502.301402023-02-24T16:48:17.026759Z"Dinh, D. C."https://zbmath.org/authors/?q=ai:dinh.doan-congSummary: Generalized Clifford algebras are constructed by various methods and have some applications in mathematics and physics. We introduce a new type of generalized Clifford algebra such that all components of a monogenic function are solutions of an elliptic partial differential equation. One of our aims is to cover more partial differential equations within the framework of Clifford analysis. We prove some Cauchy integral representation formulas for monogenic functions in these cases.Almansi-type theorem for polymonogenic functions in ballshell domainhttps://zbmath.org/1502.301412023-02-24T16:48:17.026759Z"Dinh, Doan Cong"https://zbmath.org/authors/?q=ai:dinh.doan-congSummary: The classical Almansi-type theorem for polymonogenic functions in a star-like domain in Clifford analysis is a generalization of the Almansi theorem for polyharmonic functions as well as the Fischer decomposition of polynomials. In this paper, we consider the structure of polymonogenic functions in ballshell domains. By using the Laurent series expansion of monogenic functions, we prove a new decompositions of polymonogenic functions in ballshell domains.Quaternionic views of rs-fMRI hierarchical brain activation regions. Discovery of multilevel brain activation region intensities in rs-fMRI video frameshttps://zbmath.org/1502.301422023-02-24T16:48:17.026759Z"Don, Arjuna P. H."https://zbmath.org/authors/?q=ai:don.arjuna-p-h"Peters, James F."https://zbmath.org/authors/?q=ai:peters.james-f-iii"Ramanna, Sheela"https://zbmath.org/authors/?q=ai:ramanna.sheela"Tozzi, Arturo"https://zbmath.org/authors/?q=ai:tozzi.arturoSummary: This paper introduces quaternionic views of multi-level brain activation region intensities in resting-state functional magnetic resonance imaging (rs-fMRI) videos. Quaternions make it possible to explore rs-fMRI brain activation regions in a 4D space in which there are varying brain activation intensities in spiralling activation cycles (each with its own intensity). As a result, there is a natural formation of multi-level cycles that form pyramidal vortex shapes with varying diameters. These pyramidal vortexes reflect the fractality (self-similarity) of clusters of similar multilevel brain activation region cycles. Using a computational topology of data approach, we have found that persistent, recurring \textbf{clusters of spiraling cycles} resulting from blood oxygen level dependent (BOLD) signals in triangulated rs-fMRI video frames. Each brain activation region cycle is a \textbf{cell complex}, which is a collection path-connected vertexes that has no end vertex. Measurement of persistence of spiraling vortex shapes in BOLD signal propagation regions is carried out in terms of Betti numbers (counts of distinguished cycle vertexes called generators) that rise and fall over time during spontaneous activity of the brain. A main result given here is that every quaternionic brain activation region vortex has a free group presentation. In addition, we introduce 3D barcodes of brain activation videos that help visualize and quantify the fractality of clusters of multilevel vortexes arising naturally from triangulated brain activation regions in rs-fMRI video frames. We have made freely available downloadable archives of videos that exhibit the resulting clusters of spiraling brain activation cycles.Bergman theory for the inhomogeneous Cimmino systemhttps://zbmath.org/1502.301432023-02-24T16:48:17.026759Z"González-Cervantes, José Oscar"https://zbmath.org/authors/?q=ai:gonzalez-cervantes.jose-oscar"Arroyo-Sánchez, Dante"https://zbmath.org/authors/?q=ai:arroyo-sanchez.dante"Bory-Reyes, Juan"https://zbmath.org/authors/?q=ai:bory-reyes.juanSummary: We first prove a Cauchy's integral theorem and a Cauchy-type formula for certain inhomogeneous Cimmino system from quaternionic analysis perspective. The second part of the paper directs the attention towards some applications of the mentioned results, dealing in particular with four kinds of weighted Bergman spaces, reproducing kernels, projection and conformal invariant properties.Szegö projections for Hardy spaces in quaternionic Clifford analysishttps://zbmath.org/1502.301442023-02-24T16:48:17.026759Z"He, Fuli"https://zbmath.org/authors/?q=ai:he.fuli"Huang, Song"https://zbmath.org/authors/?q=ai:huang.song"Ku, Min"https://zbmath.org/authors/?q=ai:ku.minSummary: In this paper we study Szegö kernel projections for Hardy spaces in quaternionic Clifford analysis. At first we introduce the matrix Szegö projection operator for the Hardy space of quaternionic Hermitean monogenic functions by the characterization of the matrix Hilbert transform in the quaternionic Clifford analysis. Then we establish the Kerzman-Stein formula which closely connects the matrix Szegö projection operator with the Hardy projection operator onto the Hardy space, and we get the matrix Szegö projection operator in terms of the Hardy projection operator and its adjoint. At last, we construct the explicit matrix Szegö kernel function for the Hardy space on the sphere as an example, and get the solution to a Diriclet boundary value problem for matrix functions.A note on a generalisation of Eneström-Kakeya theorem for quaternionic polynomialshttps://zbmath.org/1502.301452023-02-24T16:48:17.026759Z"Liman, Abdul"https://zbmath.org/authors/?q=ai:liman.abdul"Hussain, Shahbaz"https://zbmath.org/authors/?q=ai:hussain.shahbaz"Hussain, Imtiaz"https://zbmath.org/authors/?q=ai:hussain.imtiazSummary: In this paper, we extend some results on Eneström-Kakeya theorem which are concerned with the location of the zeros of a polynomial with quaternionic variable.On the zeros of a quaternionic polynomial with restricted coefficientshttps://zbmath.org/1502.301462023-02-24T16:48:17.026759Z"Milovanović, Gradimir V."https://zbmath.org/authors/?q=ai:milovanovic.gradimir-v"Mir, Abdullah"https://zbmath.org/authors/?q=ai:mir.abdullah"Ahmad, Abrar"https://zbmath.org/authors/?q=ai:ahmad.abrarSummary: In this paper, we are concerned with the problem of locating the zeros of regular polynomials of a quaternionic variable with restricted quaternionic coefficients. We derive new bounds of Eneström-Kakeya type for the zeros of these polynomials by virtue of a maximum modulus theorem and the structure of the zero sets established in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our results generalize some recently proved results about the distribution of zeros of a quaternionic polynomial.Extension of Grace's theorem to bi-complex polynomialshttps://zbmath.org/1502.301472023-02-24T16:48:17.026759Z"Wani, Zahid Manzoor"https://zbmath.org/authors/?q=ai:wani.zahid-manzoor"Shah, Wali Mohammad"https://zbmath.org/authors/?q=ai:shah.wali-mohammadSummary: In this paper, we prove some results concerning the zeros of Bi-complex polynomials. These results as special cases include Grace's theorem and related results.Generalized Stević-Sharma type operators from Hardy spaces into \(n\)th weighted type spaceshttps://zbmath.org/1502.301482023-02-24T16:48:17.026759Z"Abbasi, Ebrahim"https://zbmath.org/authors/?q=ai:abbasi.ebrahim"Liu, Yongmin"https://zbmath.org/authors/?q=ai:liu.yongmin"Hassanlou, Mostafa"https://zbmath.org/authors/?q=ai:hassanlou.mostafaSummary: In this paper, some characterizations for boundedness, essential norm and compactness of generalized Stevic-Sharma type operators from Hardy spaces into \(n\)th weighted type spaces are given.Nearly outer functions as extreme points in punctured Hardy spaceshttps://zbmath.org/1502.301492023-02-24T16:48:17.026759Z"Dyakonov, Konstantin M."https://zbmath.org/authors/?q=ai:dyakonov.konstantin-mLet \(H^1\) be the Hardy space, \(\{\hat{f}(k)\,/\,k\in\mathbb{Z}\}\) the sequence of Fourier coefficients of a function \(f\) in \(H^1\), \(\mathcal{K}\) a finite set of positive integers, and
\[
H_{\mathcal{K}}^1=\{f\in H^1\,/\,\hat{f}(k)=0\text{ for all }k\in\mathcal{K}\}.
\]
A classical result of \textit{K. de Leeuw} and \textit{W. Rudin} [Pac. J. Math. 8, 467--485 (1958; Zbl 0084.27503)] identifies the extreme points of the unit ball in \(H^1\) as outer functions (a point of ball(\(H^1\)) is said to be extreme if it is not the midpoint of any nondegenerate line segment contained in ball(\(H^1\))). In this very interesting paper, the author characterizes the extreme points of the ball in \(H_{\mathcal{K}}^1\) as unit-norm functions which are not too far from being outer. His main result states that \(f\) in \(H_{\mathcal{K}}^1\) with \(\| f\|_1=1\) is an extreme point of ball(\(H_{\mathcal{K}}^1\)) if and only if: (a) The inner function \(I\) in its canonical factorization is a finite Blaschke product of degree not exceeding \(\#\mathcal{K}\); (b) The rank of a certain matrix, built from the outer factor of \(f\) and the zeros of \(I\), is twice the degree of \(I\).
The author also proves that if \(f\) is an extreme point of ball(\(H_{\mathcal{K}}^1\)) and \(1/f\) is in the space \(L^1\) of integrable functions on the circle, then \(f\) is an exposed point of the ball, that is, there is a functional \(\phi\in (H_{\mathcal{K}}^1)^*\) of norm 1 such that
\[
\{g \in \text{ball}(H_{\mathcal{K}}^1)\,/\,\phi(g)=1\}=\{f\}.
\]
Reviewer: Francesc Tugores (Ourense)A Montel-type theorem for Hardy spaces of holomorphic functionshttps://zbmath.org/1502.301502023-02-24T16:48:17.026759Z"Fernández Vidal, Tomás"https://zbmath.org/authors/?q=ai:fernandez-vidal.tomas"Galicer, Daniel"https://zbmath.org/authors/?q=ai:galicer.daniel"Sevilla-Peris, Pablo"https://zbmath.org/authors/?q=ai:sevilla-peris.pabloSummary: We give a version of the Montel theorem for Hardy spaces of holomorphic functions on an infinite dimensional space. Precisely, we show that any bounded sequence of holomorphic functions in some Hardy space, has a subsequence that converges uniformly over compact subsets to a function that also belongs to the same Hardy space. As a by-product of our results for spaces of functions on infinitely many variables, we also provide an elementary proof of a Montel-type theorem for the Hardy space of Dirichlet series.Functionals with extrema at reproducing kernelshttps://zbmath.org/1502.301512023-02-24T16:48:17.026759Z"Kulikov, Aleksei"https://zbmath.org/authors/?q=ai:kulikov.alekseiSummary: We show that certain monotone functionals on the Hardy spaces and convex functionals on the Bergman spaces are maximized at the normalized reproducing kernels among the functions of norm 1, thus proving the contractivity conjecture of \textit{M. Pavlović} [Function classes on the unit disc. An introduction. Berlin: de Gruyter (2014, Zbl 1296.3002)]
and of \textit{O. Brevig} et al. [Funct. Approximatio, Comment. Math. 59, No. 1, 41-56 (2018; Zbl 1405.30054)] and the Wehrl-type entropy conjecture for the \(SU(1, 1)\) group of \textit{E. H. Lieb} and \textit{J. P. Solovej} [in: Partial differential equations, spectral theory, and mathematical physics. The Ari Laptev anniversary volume. Berlin: EMS. 301--314 (2021; 1480.81064)], respectively.
.Generic versions of a result in the theory of Hardy spaceshttps://zbmath.org/1502.301522023-02-24T16:48:17.026759Z"Nestoridis, Vassili"https://zbmath.org/authors/?q=ai:nestoridis.vassili"Thirios, Efstratios"https://zbmath.org/authors/?q=ai:thirios.efstratiosSummary: We show generic existence of functions \(f\) in the Hardy space \(H^p\) (\(0<p<1\)) on the open unit disc whose primitive \(F(f)\) satisfies the following. For every \(a>\dfrac{p}{1-p}\) and every \(A,B\in\mathbb{R}, A<B\) it holds
\[
\sup_{0<r<1}\int^B_A |F(f)(re^{i\theta})|^a d\theta =+\infty.
\]
Results of similar nature are valid when the space \(H^p\) is replaced by localized versions of it, \(0<p<1\), or intersections of such spaces.A note on Hadamard-Bergman convolution operatorshttps://zbmath.org/1502.301532023-02-24T16:48:17.026759Z"Karapetyants, Alexey"https://zbmath.org/authors/?q=ai:karapetyants.aleksei-nikolaevich"Morales, Evelyn"https://zbmath.org/authors/?q=ai:morales.evelynSummary: This paper is a continuation of the recent studies of the class of Hadamard-Bergman operators. The notion of an operator constructed according to the Carleson measure on the Bergman space is introduced, the Berezin transform of the Hadamard-Bergman operator is studied, and the sufficient conditions are given for compactness and belonging to the Schatten ideals in various cases, that is, when the kernel of the operator is a holomorphic function, an \(L^1 (\mathbb{D})\) function, or when the operator is constructed via the measure of Carleson.Generalized Libera operator on mixed-norm spaceshttps://zbmath.org/1502.301542023-02-24T16:48:17.026759Z"Naik, S."https://zbmath.org/authors/?q=ai:naik.sunanda"Rajbangshi, K."https://zbmath.org/authors/?q=ai:rajbangshi.karabiSummary: In this paper, we discuss the boundedness of generalized Libera operator \(\varLambda^{\gamma}\) on mixed-norm spaces \(H^{p,q}_{\alpha ,\nu}\). As a consequence, we find few results about the action of the operator \(\varLambda^{\gamma}\) on various function spaces such as Hardy, Zygmund, Lipschitz, Bloch type, and Besov spaces.
For the entire collection see [Zbl 1491.65006].Weakly separated Bessel systems of model spaceshttps://zbmath.org/1502.301552023-02-24T16:48:17.026759Z"Dayan, Alberto"https://zbmath.org/authors/?q=ai:dayan.albertoSummary: We show that any weakly separated Bessel system of model spaces in the Hardy space on the unit disc is a Riesz system and we highlight some applications to interpolating sequences of matrices. This will be done without using the recent solution of the Feichtinger conjecture, whose natural generalization to multidimensional model subspaces of \(\mathrm{H}^2\) turns out to be false.Some properties of Blaschke type products for the half-planehttps://zbmath.org/1502.301562023-02-24T16:48:17.026759Z"Mikaelyan, G. V."https://zbmath.org/authors/?q=ai:mikaelyan.g-v"Petrosyan, V. S."https://zbmath.org/authors/?q=ai:petrosyan.v-shSummary: In this paper we obtain balance formulas for the logarithmic means of Blaschke type functions and investigate their boundary values.Conformal transformation of uniform domains under weights that depend on distance to the boundaryhttps://zbmath.org/1502.301572023-02-24T16:48:17.026759Z"Gibara, Ryan"https://zbmath.org/authors/?q=ai:gibara.ryan"Shanmugalingam, Nageswari"https://zbmath.org/authors/?q=ai:shanmugalingam.nageswariSummary: The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain.Sphericalization and flattening in quasi-metric measure spaceshttps://zbmath.org/1502.301582023-02-24T16:48:17.026759Z"Zhou, Qingshan"https://zbmath.org/authors/?q=ai:zhou.qingshan"Li, Xining"https://zbmath.org/authors/?q=ai:li.xining"Ponnusamy, Saminathan"https://zbmath.org/authors/?q=ai:ponnusamy.saminathan"Li, Yaxiang"https://zbmath.org/authors/?q=ai:li.yaxiangSummary: The purpose of the note is to explore the invariance properties of sphericalization and flattening in quasi-metric spaces. We show that the Ahlfors regular and doubling property of quasi-metric spaces are preserved under sphericalization and flattening transformations. As an application, we give an improvement of a recent result in [\textit{X. Wang} and the first author, Ann. Acad. Sci. Fenn., Math. 42, No. 1, 257--284 (2017; Zbl 1361.30109)].Directional convexity of convolutions of harmonic functions with certain dilatationshttps://zbmath.org/1502.310032023-02-24T16:48:17.026759Z"Garg, Raj K."https://zbmath.org/authors/?q=ai:garg.raj-kumar"Dorff, Michael"https://zbmath.org/authors/?q=ai:dorff.michael-john"Jahangiri, Jay M."https://zbmath.org/authors/?q=ai:jahangiri.jay-mSummary: In an earlier paper, the second and the third author [Complex Var. Elliptic Equ. 66, No. 11, 1904--1921 (2021; Zbl 1482.31003)], considered the classes of harmonic univalent functions \(f_k=h_k+\overline{g_k}\), \(k=1,2\), that are shears of \(h_k-g_k=\frac{1}{2} \log [\left( 1+z\right) / \left( 1-z\right) ]\) with dilatations \(\omega_k=e^{i\theta_k}z^{n_k}\) for \(n_k\) positive integers, and proved that if the convolution \(f_1*f_2= h_1*h_2+\overline{g_1*g_2}\) is locally one-to-one and sense-preserving, then \(f_1*f_2\) is convex in the direction of the real axis. Later, the first and the second author [Comput. Methods Funct. Theory 22, No. 3, 519--534 (2022; Zbl 07581351)] considered \(f_m=h_m+\overline{g_m}\), \(m=1,2\), where \(h_m+e^{i \theta_m}g_m=\frac{1}{2} \log [\left( 1+z\right) / \left( 1-z\right) ]\) and determined conditions on \(\omega_m\) such that the convolution \(f_1*f_2\) is convex in the direction of the imaginary axis provided \(f_1*f_2\) is locally one-to-one and sense-preserving. In this paper, we consider the harmonic univalent functions \(f_j=h_j+\overline{g_j}\), \(j=1,2\), where \(h_1-g_1=\frac{1}{2} \log [\left( 1+z\right) / \left( 1-z\right) ]\), and we determine conditions on \(f_2\) and on \(\omega_1 = g^\prime_1/h^\prime_1\) such that the convolution \(f_1*f_2\) is typically real and convex in the direction of the imaginary axis provided that \(f_1*f_2\) is locally one-to-one and sense-preserving.Harmonic Archimedean and hyperbolic spirallikenesshttps://zbmath.org/1502.310042023-02-24T16:48:17.026759Z"Kanas, S."https://zbmath.org/authors/?q=ai:kanas.stanislawa-rSummary: We define a harmonic functions called Archimedean spirallike and hyperbolic spirallike functions. We investigate their geometric and analytic properties. Some examples are provided.Harmonic spirallike functions and harmonic strongly starlike functionshttps://zbmath.org/1502.310052023-02-24T16:48:17.026759Z"Ma, Xiu-Shuang"https://zbmath.org/authors/?q=ai:ma.xiu-shuang"Ponnusamy, Saminathan"https://zbmath.org/authors/?q=ai:ponnusamy.saminathan"Sugawa, Toshiyuki"https://zbmath.org/authors/?q=ai:sugawa.toshiyukiSummary: In this note, we define two subclasses of normalized harmonic univalent functions of the unit disk, spirallike functions and strongly starlike functions, which preserve a hereditary property and have nice analytic and geometric characterizations. We also investigate the uniform boundedness and quasiconformal extendability of strongly starlike functions. Some coefficient conditions can be given for strong starlikeness and spirallikeness. We consider a special form of harmonic functions as an application.Radius of convexity for analytic part of sense-preserving harmonic mappingshttps://zbmath.org/1502.310062023-02-24T16:48:17.026759Z"Raj, Ankur"https://zbmath.org/authors/?q=ai:raj.ankur"Nagpal, Sumit"https://zbmath.org/authors/?q=ai:nagpal.sumitSummary: Given a sense-preserving harmonic function \(f=h+\bar{g}\) defined in the open unit disk, the radius of convexity for the analytic part \(h\) is determined under various prescribed conditions on the associated analytic function \(\phi_f=h-g\). Moreover, the radius of starlikeness and convexity for the analytic part of harmonic Koebe function is also computed. All the obtained results are sharp.A note on polyharmonic mappingshttps://zbmath.org/1502.310072023-02-24T16:48:17.026759Z"Bshouty, Daoud"https://zbmath.org/authors/?q=ai:bshouty.daoud-h"Evdoridis, Stavros"https://zbmath.org/authors/?q=ai:evdoridis.stavros"Rasila, Antti"https://zbmath.org/authors/?q=ai:rasila.anttiSummary: In this paper we prove a Radó type result showing that there is no univalent polyharmonic mapping of the unit disk onto the whole complex plane. We also establish a connection between the boundary functions of harmonic and biharmonic mappings. Finally, we show how a close-to-convex biharmonic mapping can be constructed from a convex harmonic mapping.Hölder continuity of quasiconformal harmonic mappings from the unit ball to a spatial domain with \(C^1\) boundaryhttps://zbmath.org/1502.310092023-02-24T16:48:17.026759Z"Gjokaj, Anton"https://zbmath.org/authors/?q=ai:gjokaj.antonSummary: We prove that every quasiconformal mapping from the harmonic \(\beta\)-Bloch space between the unit ball and a spatial domain with \(C^1\) boundary is globally \(\alpha\)-Hölder continuous for \(\alpha < 1 - \beta\), with the Hölder coefficient that does not depend neither on the mapping nor on \(\beta\). An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for \(\alpha < 1\). This is an approach towards the extension of some results from the complex plane obtained by \textit{S. E. Warschawski} [Proc. Am. Math. Soc. 2, 254--261 (1951; Zbl 0043.08201)] for conformal mappings and \textit{D. Kalaj} [Rev. Mat. Iberoam. 38, No. 1, 95--111 (2022; Zbl 1486.30065); erratum ibid. 38, No. 1, 353--354 (2022; Zbl 1491.30010)]
for quasiconformal harmonic mappings.Statistics of square-tiled surfaces: symmetry and short loopshttps://zbmath.org/1502.320012023-02-24T16:48:17.026759Z"Shrestha, Sunrose T."https://zbmath.org/authors/?q=ai:shrestha.sunrose-t"Wang, Jane"https://zbmath.org/authors/?q=ai:wang.janeSummary: Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this article, we study counting problems that result from focusing on properties of the square torus one by one. After drawing insights from experimental evidence, we consider the implications between these properties and as well as the frequencies of these properties in each stratum of translation surfaces.Entire solutions of binomial differential equationshttps://zbmath.org/1502.341002023-02-24T16:48:17.026759Z"Gundersen, Gary G."https://zbmath.org/authors/?q=ai:gundersen.gary-g"Yang, Chung-Chun"https://zbmath.org/authors/?q=ai:yang.chung-chun\textit{W. K. Hayman} [Ann. Math. (2) 70, 9--42 (1959; Zbl 0088.28505)] has shown that if \(f\) is a meromorphic function such that \(f\) and \(f''\) have only finitely many zeros and poles, then \(f(z)=R(z)e^{P(z)}\), where \(R(z)\) is a rational function and \(P(z)\) is a polynomial, and that of these the only functions for which \(f\) and \(f''\) have no zeros are \(e^{az+b}\) and \((az + b)^{-n}\), where \(n\) is a positive integer and \(a\), \(b\) are constants. Hayman also noted that if \(f\) has no zeros, then \(f=1 / g\) where \(g\) is an entire function, and
\[
f^{\prime \prime}=\frac{2\left(g^{\prime}\right)^{2}-g g^{\prime \prime}}{g^{3}}.
\]
This leads to the question of when can the differential polynomial \(2\left(g^{\prime}\right)^{2}-g g^{\prime \prime}\) have no zeros if \(g\) is entire?
In the paper under review the authors give explicit forms of all the entire solutions of \[
f f^{\prime \prime}-a(z)\left(f^{\prime}\right)^{2}=b(z) e^{2 c(z)},
\]
where \(a(z)\), \(b(z)\), \(c(z)\) are polynomials such that \(b(z)\not\equiv 0\) and \(c(z)\) is non-constant. The paper is concluded with observations and open questions on certain binomial differential equations related to the equation above.
Reviewer: Risto Korhonen (Joensuu)Coupling of complex function theory and finite element method for crack propagation through energetic formulation: conformal mapping approach and reduction to a Riemann-Hilbert problemhttps://zbmath.org/1502.350482023-02-24T16:48:17.026759Z"Legatiuk, Dmitrii"https://zbmath.org/authors/?q=ai:legatiuk.dmitrii"Weisz-Patrault, Daniel"https://zbmath.org/authors/?q=ai:weisz-patrault.danielSummary: In this paper we present a theoretical background for a coupled analytical-numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical-numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of complex function theory and couple it continuously with the finite element solution in the region far from the singularity. In this way, crack propagation could be modelled without using remeshing. Possible directions of crack growth can be calculated through the minimization of the total energy composed of the potential energy and the dissipated energy based on the energy release rate. Within this setting, an analytical solution of a mixed boundary value problem based on complex analysis and conformal mapping techniques is presented in a circular region containing an arbitrary crack path. More precisely, the linear elastic problem is transformed into a Riemann-Hilbert problem in the unit disk for holomorphic functions. Utilising advantages of the analytical solution in the region near the crack tip, the total energy could be evaluated within short computation times for various crack kink angles and lengths leading to a potentially efficient way of computing the minimization procedure. To this end, the paper presents a general strategy of the new coupled approach for crack propagation modelling. Additionally, we also discuss obstacles in the way of practical realisation of this strategy.Monogenic functions with values in commutative algebras of the second rank with unit and the generalized biharmonic equation with double characteristichttps://zbmath.org/1502.350532023-02-24T16:48:17.026759Z"Gryshchuk, S. V."https://zbmath.org/authors/?q=ai:gryshchuk.serhii-vSummary: We prove that any commutative and associative algebra \(\mathbb{B}_*\) of the second rank with unit over the field of complex numbers \(\mathbb{C}\) contains bases \(\{e_1, e_2\}\) for which \(\mathbb{B}_*\)-valued ``analytic'' functions \(\Phi ( xe_1 + ye_2)\), where \(x\) and \(y\) are real variables, satisfy a homogeneous partial differentional equation of the fourth order with complex coefficients whose characteristic equation has a single multiple root and the remaining roots are simple. We present a complete description of the set of all triples \(( \mathbb{B}_*\), \(\{e_1, e_2\}, \Phi )\).A note on electrified dropletshttps://zbmath.org/1502.351122023-02-24T16:48:17.026759Z"Hayford, Nathan"https://zbmath.org/authors/?q=ai:hayford.nathan"Wang, Fudong"https://zbmath.org/authors/?q=ai:wang.fudongSummary: We give an in-depth analysis of a 1-parameter family of electrified droplets first described in [\textit{D. Khavinson} et al., Comput. Methods Funct. Theory 5, No. 1, 19--48 (2005; Zbl 1095.30031)]. We also investigate a technique for searching for new solutions to the droplet equation, and rederive via this technique a 1 parameter family of physical droplets, which were first discovered by \textit{D. Crowdy} [Phys. Fluids 11, No. 10, 2836--2845 (1999; Zbl 1149.76354)]. We speculate on extensions of these solutions, in particular to the case of a droplet with multiple connected components.Dynamics of hyperbolic correspondenceshttps://zbmath.org/1502.370542023-02-24T16:48:17.026759Z"Siqueira, Carlos"https://zbmath.org/authors/?q=ai:siqueira.carlosSummary: This paper establishes the geometric rigidity of certain holomorphic correspondences in the family \((w-c)^q=z^p\), whose post-critical set is finite in any bounded domain of \(\mathbb{C}\). In spite of being rigid on the sphere, such correspondences are \(J\)-stable by means of holomorphic motions when viewed as maps of \(\mathbb{C}^2\). The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.Discontinuity of straightening in anti-holomorphic dynamics. IIhttps://zbmath.org/1502.370552023-02-24T16:48:17.026759Z"Inou, Hiroyuki"https://zbmath.org/authors/?q=ai:inou.hiroyuki"Mukherjee, Sabyasachi"https://zbmath.org/authors/?q=ai:mukherjee.sabyasachiThe connected locus of the quadratic anti-holomorphic family \(f_c=\overline{z}^2+c\), \(c \in \mathbb{C}\), is called tricorn. The authors explain the occurence of tricorn-like components for real cubic rational maps, as observed by
\textit{J. Milnor} [Exp. Math. 1, No. 1, 5--24 (1992; Zbl 0762.58018)], and show that the straightening maps between these tricorn-like sets to the tricorn is not continuous.
For Part I, see [\textit{H. Inou} and \textit{S. Mukherjee}, Trans. Am. Math. Soc. 374, No. 9, 6445--6481 (2021; Zbl 1487.37057)].
Reviewer: Tao Chen (New York)S-contours and convergent interpolationhttps://zbmath.org/1502.410012023-02-24T16:48:17.026759Z"Yattselev, Maxim L."https://zbmath.org/authors/?q=ai:yattselev.maxim-lFor the entire collection see [Zbl 1478.00019].On constants in estimates of approximations by entire functions of exponential type in terms of moduli of continuity of derivativeshttps://zbmath.org/1502.410032023-02-24T16:48:17.026759Z"Babushkin, M. V."https://zbmath.org/authors/?q=ai:babushkin.max-v"Vinogradov, O. L."https://zbmath.org/authors/?q=ai:vinogradov.oleg-leonidovichSummary: We propose a new method for proving Jackson type inequalities for not necessarily periodic functions defined on the whole real line. In the inequalities under consideration, the best approximations by entire functions of exponential type are estimated in terms of the moduli of continuity of the derivatives of the approximated function. For some values of the parameters the obtained constants are less than the known ones. We construct linear approximation methods for realizing the obtained inequalities.Polynomial Hermite Padé \(m\)-system and reconstruction of the values of algebraic functionshttps://zbmath.org/1502.410052023-02-24T16:48:17.026759Z"Komlov, Aleksandr"https://zbmath.org/authors/?q=ai:komlov.aleksandr-vSummary: For an arbitrary tuple of \(m+1\) analytic germs at some fixed point, we introduce \textit{the polynomial Hermite-Padé} \(m\)-\textit{system}, which consists of \(m\) tuples of polynomials. We find weak asymptotics of such polynomials in the case where Hermite-Padé \(m\)-system is constructed by a tuple of germs of meromorphic functions on some \((m+1)\)-sheeted compact Riemann surface \(\mathfrak{R}\) under an additional condition on \(\mathfrak{R}\). As a corollary, we get a new method of reconstruction of values of an algebraic function \(f\) of order \(m+1\), determined by its initial germ \(f_0\), on all Nuttall's sheets of its Riemann surface except the last one via polynomials of Hermite-Padé \(m\)-system constructed by the tuple \([1, f_0, f_0^2,\dots ,f_0^m]\).
For the entire collection see [Zbl 1478.00019].Sharp weighted bounds for multilinear fractional type operators associated with Bergman projectionhttps://zbmath.org/1502.420152023-02-24T16:48:17.026759Z"Zhang, Juan"https://zbmath.org/authors/?q=ai:zhang.juan.4|zhang.juan.3|zhang.juan.1|zhang.juan.2"Lan, Senhua"https://zbmath.org/authors/?q=ai:lan.senhua"Xue, Qingying"https://zbmath.org/authors/?q=ai:xue.qingyingSummary: We first introduce the multiple weights which are suitable for the study of Bergman type operators. Then, we give the sharp weighted estimates for multilinear fractional Bergman operators and fractional maximal function.Affine density, von Neumann dimension and a problem of Perelomovhttps://zbmath.org/1502.420252023-02-24T16:48:17.026759Z"Abreu, Luís Daniel"https://zbmath.org/authors/?q=ai:abreu.luis-daniel"Speckbacher, Michael"https://zbmath.org/authors/?q=ai:speckbacher.michaelAuthors' abstract: We provide a solution to Perelomov's 1972 problem [\textit{A. M. Perelomov}, Funct. Anal. Appl. 7, 215--222 (1974; Zbl 0295.22021); translation from Funkts. Anal. Prilozh. 7, No. 3, 57--66 (1973)] concerning the existence of a phase transition (known in signal analysis as `Nyquist rate') determining the basis properties of certain affine coherent states labelled by Fuchsian groups. As suggested by Perelomov, the transition is given according to the hyperbolic volume of the fundamental region. The solution is a more general form (in phase space) of the \(\mathrm{PSL}(2, \mathbb{R})\) variant of a 1989 conjecture of \textit{K. Seip} [``Mean value theorems and concentration operators in Bargmann and Bergman space'', in: Wavelets. Inverse problems and theoretical imaging. Berlin, Heidelberg: Springer. 209--215 (1989; \url{https://doi.org/10.1007/978-3-642-97177-8_18})] about wavelet frames, where the same value of `Nyquist rate' is obtained as the trace of a certain localization operator. The proof consists of first connecting the problem to the theory of von Neumann algebras, by introducing a new class of projective representations of \(\mathrm{PSL}(2, \mathbb{R})\) acting on non-analytic Bergman-type spaces. We then adapt to this setting a new method for computing von Neumann dimensions, due to Sir Vaughan Jones. Our solution contains necessary conditions in the form of a `Nyquist rate' dividing frames from Riesz sequences of coherent states and sampling from interpolating sequences. They hold for an infinite sequence of spaces of polyanalytic functions containing the eigenspaces of the Maass operator and their orthogonal sums. Within mild boundaries, we show that our result is best possible, by characterizing our sequence of function spaces as the only invariant spaces under the non-analytic \(\mathrm{PSL}(2, \mathbb{R})\)-representations.
Reviewer: Jakob Lemvig (Kgs. Lyngby)A covariance equationhttps://zbmath.org/1502.430042023-02-24T16:48:17.026759Z"Youssfi, El Hassan"https://zbmath.org/authors/?q=ai:youssfi.el-hassanSummary: Let \(\mathbb{S}\) be a commutative semigroup with identity \(e\) and let \(\varGamma\) be a compact subset in the pointwise convergence topology of the space \(\mathbb{S}'\) of all non-zero multiplicative functions on \(\mathbb{S}\). Given a continuous function \(F:\varGamma\rightarrow \mathbb{C}\) and a complex regular Borel measure \(\mu\) on \(\varGamma\) such that \(\mu(\varGamma)\neq 0\). It is shown that
\[
\mu(\varGamma)\int_{\varGamma}\varrho(s)\overline{\varrho(t)}|F|^2(\varrho) \,\text{d}\mu(\varrho)=\int_{\varGamma}\varrho (s)F(\varrho)\,\text{d}\mu(\varrho) \int_{\varGamma}\overline{\varrho (t) F(\varrho)}\,\text{d}\mu (\varrho)
\]
for all \((s,t)\in\mathbb{S}\times\mathbb{S}\) if and only if for some \(\gamma\in\varGamma\), the support of \(\mu\) is contained in \(\{F=0\}\cup\{\gamma\}\). Several applications of this characterization are derived. In particular, the reduction of our theorem to the semigroup of non-negative integers \((\mathbb{N}_0,+)\) solves a problem posed by El Fallah, Klaja, Kellay, Mashregui, and Ransford [3] in a more general context. More consequences of our results are given, some of them illustrate the probabilistic flavor behind the problem studied herein and others establish extremal properties of analytic kernels.On dual transform of fractional Hankel transformhttps://zbmath.org/1502.440032023-02-24T16:48:17.026759Z"Ghanmi, Allal"https://zbmath.org/authors/?q=ai:ghanmi.allalSummary: We deal with a class of one-parameter family of integral transforms of Bargmann type arising as dual transform of fractional Hankel transform. Their ranges are identified to be special subspaces of the weighted hyperholomorphic left Hilbert spaces, generalizing the slice Bergman space of the second kind. Their reproducing kernel is given by closed expression involving the \(\star \)-regularization of Gauss hypergeometric function. We also discuss their basic properties such as boundedness and we determine their singular values. Moreover, we describe their compactness and membership in \(p\)-Schatten classes. The operational calculus for this transform is also investigated.Hirschman-Widder densitieshttps://zbmath.org/1502.440042023-02-24T16:48:17.026759Z"Belton, Alexander"https://zbmath.org/authors/?q=ai:belton.alexander-c-r"Guillot, Dominique"https://zbmath.org/authors/?q=ai:guillot.dominique"Khare, Apoorva"https://zbmath.org/authors/?q=ai:khare.apoorva"Putinar, Mihai"https://zbmath.org/authors/?q=ai:putinar.mihaiSummary: Hirschman and Widder introduced a class of Pólya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the class is closed under convolution but not under pointwise multiplication. We show that, generically, a polynomial function of such a density is a Pólya frequency function only if the polynomial is a homothety, and also identify a subclass for which each positive-integer power is a Pólya frequency function. We further demonstrate connections between the Maclaurin coefficients, the moments of these densities, and the recovery of the density from finitely many moments, via Schur polynomials.Holomorphic solutions and solvability theory for a class of linear complete singular integro-differential equations with convolution by Riemann-Hilbert methodhttps://zbmath.org/1502.450032023-02-24T16:48:17.026759Z"Li, Pingrun"https://zbmath.org/authors/?q=ai:li.pingrunSummary: In this article, we investigate a class of linear complete singular integro-differential equations of convolution type with Fredholm integral operator in the function class \(\{0\}\). To prove the existence of solution for the equations, we first propose the regularization method of complex integral equations, and then we establish the theory of Noether solvability. By using the relation between Cauchy type integral and Fourier integral, we transform such equations into the complete singular integral equations, and on this basis we transform further the obtained equations into boundary value problems of holomorphic functions with node (i.e., discontinuous coefficients). The holomorphic solution and conditions of solvability are obtained via the method of regularization. Our approach of solving equations is novel and effective, which is different from the classical method. Meanwhile, we also discuss the asymptotic stability of solutions. Therefore, this article has a great significance for the study of developing functional analysis, complex analysis, and differential-integral equations, also provides theoretical support for the study of engineering mechanics, elastic mechanics and mathematical Modeling.Reducibility for a class of analytic multipliers on Sobolev disk algebrahttps://zbmath.org/1502.470442023-02-24T16:48:17.026759Z"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yong.4"Liu, Ya"https://zbmath.org/authors/?q=ai:liu.ya"Qin, Chuntao"https://zbmath.org/authors/?q=ai:qin.chuntaoSummary: We prove the reducibility of analytic multipliers \(M_\phi\) with a class of finite Blaschke products symbol \(\phi\) on the Sobolev disk algebra \(R(\mathbb{D})\). We also describe their nontrivial minimal reducing subspaces.The product-type operators from the Besov spaces into \(n\)th weighted type spaceshttps://zbmath.org/1502.470482023-02-24T16:48:17.026759Z"Abbasi, Ebrahim"https://zbmath.org/authors/?q=ai:abbasi.ebrahimSummary: The main goal of this paper is to investigate of boundedness and compactness of a class of product-type operators \(T^m_{u,v,\varphi}\) from Besov spaces into \(n\)th weighted type spaces.Essential norms of Volterra and Cesàro operators on Müntz spaceshttps://zbmath.org/1502.470492023-02-24T16:48:17.026759Z"Alam, Ihab Al"https://zbmath.org/authors/?q=ai:alam.ihab-al"Habib, Georges"https://zbmath.org/authors/?q=ai:habib.georges"Lefèvre, Pascal"https://zbmath.org/authors/?q=ai:lefevre.pascal"Maalouf, Fares"https://zbmath.org/authors/?q=ai:maalouf.faresSummary: We study the properties of the Volterra and Cesàro operators viewed on the \(L^1\)-Müntz space \(M_\varLambda ^1\) with range in the space of continuous functions. These operators are neither compact nor weakly compact. We estimate how far they are from being (weakly) compact by computing their (generalized) essential norm. It turns out that this norm does not depend on \(\varLambda \) and is equal to \(1/2\).Truncations of random unitary matrices drawn from Hua-Pickrell distributionhttps://zbmath.org/1502.600072023-02-24T16:48:17.026759Z"Lin, Zhaofeng"https://zbmath.org/authors/?q=ai:lin.zhaofeng"Qiu, Yanqi"https://zbmath.org/authors/?q=ai:qiu.yanqi"Wang, Kai"https://zbmath.org/authors/?q=ai:wang.kai|wang.kai.5|wang.kai.4|wang.kai.2|wang.kai.3Summary: Let \(U\) be a random unitary matrix drawn from the Hua-Pickrell distribution \(\mu_{\mathrm{U}(n+m)}^{(\delta)}\) on the unitary group \(\mathrm{U}(n+m)\). We show that the eigenvalues of the truncated unitary matrix \([U_{i, j}]_{1\leq i, j\leq n}\) form a determinantal point process \(\mathscr{X}_n^{(m, \delta)}\) on the unit disc \(\mathbb{D}\) for any \(\delta\in\mathbb{C}\) satisfying \(\Re\delta > -1/2\). We also prove that the limiting point process taken by \(n\rightarrow\infty\) of the determinantal point process \(\mathscr{X}_n^{(m, \delta)}\) is always \(\mathscr{X}^{[m]}\), independent of \(\delta\). Here \(\mathscr{X}^{[m]}\) is the determinantal point process on \(\mathbb{D}\) with weighted Bergman kernel
\[
K^{[m]}(z,w) = \frac{1}{(1-z{\overline{w}})^{m+1}}
\]
with respect to the reference measure \(d\mu^{[m]}(z) = \frac{m}{\pi}(1 - |z|)^{m-1}d\sigma(z)\), where \(d\sigma(z)\) is the Lebesgue measure on \(\mathbb{D}\).The fundamental inequality for cocompact Fuchsian groupshttps://zbmath.org/1502.600602023-02-24T16:48:17.026759Z"Kosenko, Petr"https://zbmath.org/authors/?q=ai:kosenko.petr"Tiozzo, Giulio"https://zbmath.org/authors/?q=ai:tiozzo.giulioSummary: We prove that the hitting measure is singular with respect to the Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups.Effective mass theorems with Bloch modes crossingshttps://zbmath.org/1502.810322023-02-24T16:48:17.026759Z"Chabu, Victor"https://zbmath.org/authors/?q=ai:chabu.victor"Fermanian Kammerer, Clotilde"https://zbmath.org/authors/?q=ai:fermanian-kammerer.clotilde"Macià, Fabricio"https://zbmath.org/authors/?q=ai:macia.fabricioSummary: We study a Schrödinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wave-length comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make a rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.Analysis of unitarity in conformal quantum gravityhttps://zbmath.org/1502.830042023-02-24T16:48:17.026759Z"Kubo, Jisuke"https://zbmath.org/authors/?q=ai:kubo.jisuke"Kuntz, Jeffrey"https://zbmath.org/authors/?q=ai:kuntz.jeffreySummary: We perform a canonical quantization of Weyl's conformal gravity by means of the covariant operator formalism and investigate the unitarity of the resulting quantum theory. After reducing the originally fourth-order theory to second-order in time derivatives via the introduction of an auxiliary tensor field, we identify the full Fock space of quantum states under a Becchi-Rouet-Stora-Tyutin (BRST) construction that includes Faddeev-Popov ghost fields corresponding to Weyl transformations. This second-order formulation allows the formal tools of operator-based quantum field theory to be applied to quadratic gravity for the first time. Using the Kugo-Ojima quartet mechanism, we identify the physical subspace of quantum states and find that the subspace containing the transverse spin-2 states comes equipped with an indefinite inner product metric and a one-particle Hamiltonian that possesses only a single eigenstate. We construct the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula for the \(S\)-matrix in this spin-2 subspace and find that unitarity is violated in scattering events. The explicit way in which this violation occurs represents a new view on the ghost-problem in quadratic theories of quantum gravity.