Recent zbMATH articles in MSC 30Chttps://zbmath.org/atom/cc/30C2024-02-15T19:53:11.284213ZWerkzeugGeneralizations of some differential inequalities for polynomialshttps://zbmath.org/1526.300012024-02-15T19:53:11.284213Z"Mir, M. Y."https://zbmath.org/authors/?q=ai:mir.mohd-yousf"Wali, S. L."https://zbmath.org/authors/?q=ai:wali.shah-lubn"Shah, W. M."https://zbmath.org/authors/?q=ai:shah.wali-mohammadSummary: We consider polynomials of the form \(P(z)=z^s\left(a_0+\sum_{v=t}^{n-s}a_vz^v\right)\), \(t\geq 1\), \(0\leq s\leq n-1\) and prove some results for the estimate of the polar derivative \(D_{\alpha}P(z):=nP(z)+(\alpha-z)P^{\prime}(z)\) and generalize the results due to \textit{A. Aziz} and the third author [Proc. Indian Acad. Sci., Math. Sci. 107, No. 3, 263--270 (1997; Zbl 0898.41009)], \textit{N. K. Govil} [J. Approximation Theory 66, No. 1, 29--35 (1991; Zbl 0735.41006)] and others.A Jentzsch-theorem for Kapteyn, Neumann and general Dirichlet serieshttps://zbmath.org/1526.300042024-02-15T19:53:11.284213Z"Bornemann, Folkmar"https://zbmath.org/authors/?q=ai:bornemann.folkmar-aSummary: Comparing phase plots of truncated series solutions of Kepler's equation by Lagrange's power series with those by Bessel's Kapteyn series strongly suggests that a Jentzsch-type theorem holds true not only for the former but also for the latter series: each point of the boundary of the domain of convergence in the complex plane is a cluster point of zeros of sections of the series. We prove this result by studying properties of the growth function of a sequence of entire functions. For series, this growth function is computable in terms of the convergence abscissa of an associated general Dirichlet series. The proof then extends, besides including Jentzsch's classical result for power series, to general Dirichlet series, to Kapteyn, and to Neumann series of Bessel functions. Moreover, sections of Kapteyn and Neumann series generally exhibit zeros close to the real axis which can be explained, including their asymptotic linear density, by the theory of the distribution of zeros of entire functions.Inequalities for the derivatives of rational functions with prescribed poles and restricted zeroshttps://zbmath.org/1526.300082024-02-15T19:53:11.284213Z"Ahanger, U. M."https://zbmath.org/authors/?q=ai:ahanger.uzma-mubeen"Shah, W. M."https://zbmath.org/authors/?q=ai:shah.wali-mohammadSummary: As a generalization of a result obtained by \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)], \textit{S. L. Wali} [``Inequalities for maximum modulus of rational functions with prescribed poles'', Preprint, Kragujevac J. Math. 47, 865--875 (2023)] recently proved the following: Let \(r \in \mathcal{R}_n \), where \(r\) has \(n\) poles at \(a_1, a_2, \dots, a_n\) and all its zeros lie in \(|z| \leq 1\), with \(s\)-fold zeros at the origin, then for \(|z| = 1\)
\[\left| {r'(z)} \right| \geqslant \frac{1}{2}\left\{ {\left| {\mathcal{B}'(z)} \right| + (s + m - n) + \frac{{\left| {{{c}_m}} \right| - \left| {{{c}_s}} \right|}}{{\left| {{{c}_m}} \right| + \left| {{{c}_s}} \right|}}} \right\}\left| {r(z)} \right|.\]
In this paper, instead of assuming that \(r(z)\) has a zero of order \(s\) at the origin as Wali did, we suppose that \(r(z)\) has a zero of multiplicity \(s\) at any point inside the unit circle and all other zeros are inside or outside a circle of radius \(k\). Further, we prove some results which besides generalizing some inequalities for rational functions include refinements of some polynomial inequalities as special cases.On the transfinite diameters of related sets. An extension of Robinson's theoremhttps://zbmath.org/1526.300092024-02-15T19:53:11.284213Z"Babayan, N. M."https://zbmath.org/authors/?q=ai:babayan.nikolay-m"Ginovyan, M. S."https://zbmath.org/authors/?q=ai:ginovyan.mamikon-sSummary: The paper is devoted to the transfinite diameters of some related sets. We discuss the Fekete theorem and its extension due to Robinson on the transfinite diameters of related sets, and prove an extension of Robinson's theorem. The transfinite diameters of some special subsets of the unit circle are calculated explicitly by using Fekete's and Robinson's theorems.A simple proof of the Fundamental Theorem of Algebrahttps://zbmath.org/1526.300102024-02-15T19:53:11.284213Z"Chakraborty, Bikash"https://zbmath.org/authors/?q=ai:chakraborty.bikashSummary: Many proofs of the Fundamental Theorem of Algebra, including various proofs based on the theory of analytic functions of a complex variable, are known. To the best of our knowledge, this proof is different from the existing ones.Strong asymptotics of planar orthogonal polynomials: Gaussian weight perturbed by finite number of point chargeshttps://zbmath.org/1526.300112024-02-15T19:53:11.284213Z"Lee, Seung-Yeop"https://zbmath.org/authors/?q=ai:lee.seung-yeop"Yang, Meng"https://zbmath.org/authors/?q=ai:yang.mengSummary: We consider the orthogonal polynomial \(p_n (z)\) with respect to the planar measure supported on the whole complex plane
\[
{{\mathrm{e}}}^{ - N|z{|}^2}\prod\limits_{j = 1}^\nu{|z - a_j{|}^{2{c}_j}} {\mathrm{d}}A(z)
\]
where \(\mathrm{d}A\) is the Lebesgue measure of the plane, \(N\) is a positive constant, \(\{c_1, \dots, c_\nu\}\) are nonzero real numbers greater than \(-1\) and \(\{ a_1, \ldots ,a_\nu \} \subset \mathbb{D}\backslash \{ 0\}\) are distinct points inside the unit disk. In the scaling limit when \(n/N = 1\) and \(n \rightarrow \infty\) we obtain the strong asymptotics of the polynomial \(p_n(z)\). We show that the support of the roots converges to what we call the ``multiple Szegő curve,'' a certain connected curve having \(\nu + 1\) components in its complement. We apply the nonlinear steepest descent method [\textit{P. Deift},
Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Reprint of the 1998 original.
Courant Lecture Notes in Mathematics 3. New York, NY: Courant Institute of Mathematical Sciences. Providence, RI: AMS (2000; Zbl 0997.47033); \textit{P. Deift} et al.,
Commun. Pure Appl. Math. 52, No. 12, 1491--1552 (1999; Zbl 1026.42024)] on the matrix Riemann-Hilbert problem of size \((\nu + 1) \times (\nu + 1)\) posed in [the authors, J. Phys. A, Math. Theor. 52, No. 27, Article ID 275202, 14 p. (2019; Zbl 1509.33010)].
{{\copyright} 2023 Wiley Periodicals, LLC.}Extremal problems for a polynomial and its polar derivativehttps://zbmath.org/1526.300122024-02-15T19:53:11.284213Z"Mir, Abdullah"https://zbmath.org/authors/?q=ai:mir.abdullahSummary: This paper considers the well known Erdős-Lax and Turán-type inequalities that relate the uniform norm of a univariate complex coefficient polynomial to that of its derivative on the unit circle in the plane. Here, we establish some new inequalities that relate the uniform norm of a polynomial and its polar derivative while taking into account the placement of the zeros and the extremal coefficients of the polynomial. The obtained results strengthen some recently proved Erdős-Lax and Turán-type inequalities for constrained polynomials and also produce various inequalities that are sharper than the previous ones known in the literature on this subject.Inequalities for rational functions with prescribed poleshttps://zbmath.org/1526.300132024-02-15T19:53:11.284213Z"Rather, N. A."https://zbmath.org/authors/?q=ai:rather.nisar-ahmad"Iqbal, A."https://zbmath.org/authors/?q=ai:iqbal.ayesha|iqbal.aklad|iqbal.adnan|iqbal.arshad|iqbal.anam|iqbal.akhlad|iqbal.adam-s|iqbal.anjum|iqbal.afshan|iqbal.asif|iqbal.azmat|iqbal.amer|iqbal.atif|iqbal.azhar|iqbal.azhar.1|iqbal.azeem"Dar, Ishfaq"https://zbmath.org/authors/?q=ai:dar.ishfaq-ahmadSummary: For rational functions \(R(z)=P(z)/W(z)\), where \(P\) is a polynomial of degree at the most \(n\) and \(W(z)=\prod_{j=1}^n(z-a_j)\), with \(|a_j|>1\), \(j\in \{1,2,\dots,n\},\) we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdős-Lax and Turán to rational functions \(R\). In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdős-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we will demonstrate how some recently obtained results could have been proved without invoking the results of \textit{V. N. Dubinin} [Sb. Math. 191, No. 12, 1797--1807 (2000); translation from Mat. Sb. 191, No. 12, 51--60 (2000; Zbl 1010.30004)].On the zeros of polynomials with restricted coefficientshttps://zbmath.org/1526.300142024-02-15T19:53:11.284213Z"Zargar, B. A."https://zbmath.org/authors/?q=ai:zargar.bashir-ahmad"Gulzar, M. H."https://zbmath.org/authors/?q=ai:gulzar.manzoor-hussain"Ali, M."https://zbmath.org/authors/?q=ai:ali.majid|ali.masoom|ali.mai|ali.mehreen|ali.madad|ali.murtuza|ali.masooma|ali.musavvir|ali.mahbub|ali.mageed|ali.muddasir|ali.mazen|ali.mehboob|ali.musrrat|ali.mohsin|ali.mabruka|ali.mchirgui|ali.mubasher|ali.muzaher|ali.maifuz|ali.musawir|ali.mutahir|ali.muqadar|ali.markus|ali.mahvish|ali.mukarram|ali.mouhib|ali.musharraf|ali.mumtaz|ali.murtaza|ali.mokshed|ali.mashkoor|ali.mohammod|ali.muktar|ali.maghari|ali.mehdi|ali.moonis|ali.mohabbat|ali.maaruf|ali.mushtaq|ali.mortuza|ali.mazhar|ali.maher|ali.muqeetSummary: Let \(P(z) = \sum\nolimits_{j = 0}^n a_j z^j\) be a polynomial of degree \(n\) such that \(a_n \geq a_{n-1} \geq \ldots \geq a_1 \geq a_0 \geq 0\). Then according to Eneström-Kakeya theorem all the zeros of \(P(z)\) lie in \(|z|\leq 1\). This result has been generalized in various ways (see [Zbl 0867.30004; Zbl 0808.30004; Zbl 0194.10201; Zbl 0162.37101; Zbl 1350.30009]. In this paper we shall prove some generalizations of the results due to \textit{A. Aziz} and the first author [Zbl 0867.30004; ``Bounds for the zeros of a polynomial with restricted coefficients'', Appl. Math.
(Irvine) 3, no. 1, 30--33 (2012)] and \textit{E. R. Nwaeze} [Zbl 1350.30009].On the quasi-stability criteria of monic matrix polynomialshttps://zbmath.org/1526.300152024-02-15T19:53:11.284213Z"Zhan, Xuzhou"https://zbmath.org/authors/?q=ai:zhan.xuzhou"Ban, Bohui"https://zbmath.org/authors/?q=ai:ban.bohui"Hu, Yongjian"https://zbmath.org/authors/?q=ai:hu.yongjianSummary: This paper is a continuation of a recent investigation by the first author and \textit{A. Dyachenko} [J. Comput. Appl. Math. 383, Article ID 113113, 16 p. (2021; Zbl 1460.34069)] on the Hurwitz stability of monic matrix polynomials with algebraic techniques. By improving an inertia formula for matrix polynomials with respect to the imaginary axis, we show that, under some conditions, the quasi-stability of a monic matrix polynomial can be tested via the Hermitian nonnegative definiteness of two block Hankel matrices built from its matricial Markov parameters. Moreover, for the so-called doubly monic matrix polynomials, the quasi-stability criteria can be formulated in a much simpler form. In particular, the relationship between Hurwitz stable matrix polynomials and Stieltjes positive definite matrix sequences established in [loc. cit.] is included as a special case.The Ptolemy-Alhazen problem and quadric surface mirror reflectionhttps://zbmath.org/1526.300162024-02-15T19:53:11.284213Z"Fujimura, Masayo"https://zbmath.org/authors/?q=ai:fujimura.masayo"Mocanu, Marcelina"https://zbmath.org/authors/?q=ai:mocanu.marcelina"Vuorinen, Matti"https://zbmath.org/authors/?q=ai:vuorinen.matti-kSummary: We discuss the problem of the reflection of light on spherical and quadric surface mirrors. In the case of spherical mirrors, this problem is known as the Alhazen problem. For the spherical mirror problem, we focus on the reflection property of an ellipse and show that the catacaustic curve of the unit circle follows naturally from the equation obtained from the reflection property of an ellipse. Moreover, we provide an algebraic equation that solves Alhazen's problem for quadric surface mirrors.Geometric versions of Schwarz's lemma for spherically convex functionshttps://zbmath.org/1526.300172024-02-15T19:53:11.284213Z"Kourou, Maria"https://zbmath.org/authors/?q=ai:kourou.maria"Roth, Oliver"https://zbmath.org/authors/?q=ai:roth.oliverSummary: We prove several sharp distortion and monotonicity theorems for spherically convex functions defined on the unit disk involving geometric quantities such as spherical length, spherical area, and total spherical curvature. These results can be viewed as geometric variants of the classical Schwarz lemma for spherically convex functions.Analogue of Kellogg's theorem for piecewise Lyapunov domainshttps://zbmath.org/1526.300182024-02-15T19:53:11.284213Z"Soldatov, A. P."https://zbmath.org/authors/?q=ai:soldatov.aleksandr-pavlovichSummary: In weighted Hölder spaces, classes of smooth arcs and piecewise smooth contours are introduced that are invariant under power mappings. The boundary properties of conformal mappings are described in terms of these classes by analogy with Kellogg's classical theorem.Lipschitz estimates for conformal maps from the unit disk to convex domainshttps://zbmath.org/1526.300192024-02-15T19:53:11.284213Z"Donohue, Christopher G."https://zbmath.org/authors/?q=ai:donohue.christopher-gSummary: We obtain an explicit uniform upper bound for the derivative of a conformal mapping of the unit disk onto a convex domain. This estimate depends only on the outer and inner radii of the domain, and on the minimum curvature radius of its boundary. Its proof is based on a Möbius invariant metric of hyperbolic type, introduced by \textit{R. S. Kulkarni} and \textit{U. Pinkall} [Math. Z. 216, No. 1, 89--129 (1994; Zbl 0813.53022)].Geometric characterizations for conformal mappings in weighted Bergman spaceshttps://zbmath.org/1526.300202024-02-15T19:53:11.284213Z"Karafyllia, Christina"https://zbmath.org/authors/?q=ai:karafyllia.christina"Karamanlis, Nikolaos"https://zbmath.org/authors/?q=ai:karamanlis.nikolaosSummary: We prove that a conformal mapping defined on the unit disk belongs to a weighted Bergman space if and only if certain integrals involving the harmonic measure converge. With the aid of this theorem, we give a geometric characterization of conformal mappings in Hardy or weighted Bergman spaces by studying Euclidean areas. Applying these results, we prove several consequences for such mappings that extend known results for Hardy spaces to weighted Bergman spaces. Moreover, we introduce a number which is the analogue of the Hardy number for weighted Bergman spaces. We derive various expressions for this number and hence we establish new results for the Hardy number and the relation between Hardy and weighted Bergman spaces.Paatero's classes \(V(k)\) as subsets of the Hornich spacehttps://zbmath.org/1526.300212024-02-15T19:53:11.284213Z"Andreev, Valentin V."https://zbmath.org/authors/?q=ai:andreev.valentin-v"Bekker, Miron B."https://zbmath.org/authors/?q=ai:bekker.miron-b"Cima, Joseph A."https://zbmath.org/authors/?q=ai:cima.joseph-aSummary: In this article we consider Paatero's classes \(V(k)\) of functions of bounded boundary rotation as subsets of the Hornich space \(\mathcal{H}\). We show that for a fixed \(k\ge 2\) the set \(V(k)\) is a closed and convex subset of \(\mathcal{H}\) and is not compact. We identify the extreme points of \(V(k)\) in \(\mathcal{H}\).On normalized rabotnov function associated with certain subclasses of analytic functionshttps://zbmath.org/1526.300222024-02-15T19:53:11.284213Z"Eker, Sevtap Sümer"https://zbmath.org/authors/?q=ai:eker.sevtap-sumer"Şeker, Bilal"https://zbmath.org/authors/?q=ai:seker.bilal"Ece, Sadettin"https://zbmath.org/authors/?q=ai:ece.sadettinSummary: In this paper, we investigate some sufficient conditions for the normalized Rabotnov function to be in certain subclasses of analytic and univalent functions. The usefulness of the results is depicted by some corollaries and examples.The third-order Hermitian Toeplitz determinant for some classes of analytic functionshttps://zbmath.org/1526.300232024-02-15T19:53:11.284213Z"Lecko, A."https://zbmath.org/authors/?q=ai:lecko.adam"Śmiarowska, Barbara"https://zbmath.org/authors/?q=ai:smiarowska.barbaraSummary: Sharp lower and upper bounds are found of the second and third-order Hermitian Toeplitz determinants for some classes of analytic functions.Sharp bounds on the Hankel determinant of the inverse functions for certain analytic functionshttps://zbmath.org/1526.300242024-02-15T19:53:11.284213Z"Shi, Lei"https://zbmath.org/authors/?q=ai:shi.lei.3|shi.lei.2|shi.lei.4"Arif, Muhammad"https://zbmath.org/authors/?q=ai:arif.muhammad"Srivastava, H. M."https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Ihsan, Muhammad"https://zbmath.org/authors/?q=ai:ihsan.muhammadSummary: In most cases, the problem of finding bounds for the inverse function is much more difficult than finding bounds for the function itself. Thus, there are relatively little sharp bounds of Hankel determinant on the inverse functions. In the present paper, we introduce a subclass of bounded turning function \(\mathscr{R}_{car}\) associated with a cardioid domain. The purpose of this article is to investigate certain coefficient related problems on the inverse functions for \(f\in\mathscr{R}_{car}\). The bounds of some initial coefficients, the Fekete-Szegö type inequality and the estimation of Hankel determinants of second and third order are obtained. All of these bounds are proved to be sharp.Sharp bounds on the fourth-order Hermitian Toeplitz determinant for starlike functions of order 1/2https://zbmath.org/1526.300252024-02-15T19:53:11.284213Z"Sun, Yong"https://zbmath.org/authors/?q=ai:sun.yong"Wang, Zhi-Gang"https://zbmath.org/authors/?q=ai:wang.zhigang"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huoSummary: In this paper, we prove the sharp bounds on the fourth-order Hermitian Toeplitz determinant for starlike functions of order 1/2, which solves a conjecture posed by \textit{K. Cudna} et al. [Bol. Soc. Mat. Mex., III. Ser. 26, No. 2, 361--375 (2020; Zbl 1435.30044)] for the case \(q = 4\).Weakly quasisymmetric mappings and uniform domains on generalized Grushin planeshttps://zbmath.org/1526.300262024-02-15T19:53:11.284213Z"Liu, Hongjun"https://zbmath.org/authors/?q=ai:liu.hongjunSummary: In this paper, we mainly consider the weakly quasisymmetric mappings and uniform domains on the generalized Grushin planes. We prove that \(\Phi : G_r \to \mathbf{C}\) is a weakly \((h, H)\)-quasisymmetric mapping. Meanwhile, we also show the invariance of uniform domains on the generalized Grushin planes.Strongly symmetric homeomorphisms on the real line with uniform continuityhttps://zbmath.org/1526.300272024-02-15T19:53:11.284213Z"Wei, Huaying"https://zbmath.org/authors/?q=ai:wei.huaying"Matsuzaki, Katsuhiko"https://zbmath.org/authors/?q=ai:matsuzaki.katsuhiko.1|matsuzaki.katsuhikoSummary: We investigate strongly symmetric homeomorphisms of the real line which appear in harmonic analysis aspects of quasiconformal Teichmüller theory. An element in this class can be characterized by a property where it can be extended quasiconformally to the upper half-plane so that its complex dilatation induces a vanishing Carleson measure. However, unlike the case on the unit circle, strongly symmetric homeomorphisms on the real line are not preserved under either the composition or the inversion. In this paper, we present the difference and the relation between these two cases. In particular, we show that if uniform continuity is assumed for strongly symmetric homeomorphisms of the real line, then they are preserved by those operations. We also show that the barycentric extension of uniformly continuous one induces a vanishing Carleson measure, as do the composition and inverse of those quasiconformal homeomorphisms of the upper half-plane.Quasisymmetry and solidity of quasiconformal maps in metric spaceshttps://zbmath.org/1526.300282024-02-15T19:53:11.284213Z"Cheng, Tao"https://zbmath.org/authors/?q=ai:cheng.tao"Jiang, Peng"https://zbmath.org/authors/?q=ai:jiang.peng"Yang, Shanshuang"https://zbmath.org/authors/?q=ai:yang.shanshuangSummary: This paper is devoted to the study of the broad problem of deriving global metric properties from local quasiconformality in metric spaces. In particular, we show that, under certain regularity and connectivity conditions, a quasiconformal map between metric spaces is \textit{weakly \((L, M)\)-quasisymmetric}. Furthermore, such a map is \textit{solid} if the metric spaces are complete. These extend and generalize some well known results in Euclidean as well as metric spaces.Boundary behaviour of open, light mappings in metric measure spaceshttps://zbmath.org/1526.300292024-02-15T19:53:11.284213Z"Cristea, Mihai"https://zbmath.org/authors/?q=ai:cristea.mihai.1|cristea.mihaiThe article is devoted to the study of mappings of metric spaces. The author considers mappings that satisfy a Poletsky-type weight inequality. The properties of the mappings are largely determined by the properties of the corresponding weight. Therefore, the author introduces a certain function that controls the behavior of this weight and assumes that the weight \(p\)-modulus of the family of paths passing through the point is equal to zero. Under these conditions, the author formulates and proves several results concerning the local and boundary behavior. In particular, he obtains results on the continuous extension of mappings to a point of a totally disconnected set, a theorem on the normality (equicontinuity) of families of such mappings, gives a partial description of the cluster set of mappings and the set of their asymptotic limits.
Reviewer: Evgeny Sevost'yanov (Zhitomir)Growth inequalities for quasimeromorphic mappings with applicationshttps://zbmath.org/1526.300302024-02-15T19:53:11.284213Z"Yang, Xuxin"https://zbmath.org/authors/?q=ai:yang.xuxin"Rasila, Antti"https://zbmath.org/authors/?q=ai:rasila.anttiSummary: We prove a version of the Picard-Schottky theorem involving the multiplicity for quasiregular in \(n\)-dimensional Euclidean space \(\mathbb{R}^n\). We also study the boundary behavior of quasiregular mappings from the unit ball \(\mathbf{B}^n\) into a multiply punctured space \(\mathbb{R}^n \setminus \{w_1, \dots, w_{p - 1}\}\).Quasihyperbolic geodesics are cone arcshttps://zbmath.org/1526.300312024-02-15T19:53:11.284213Z"Zhou, Qingshan"https://zbmath.org/authors/?q=ai:zhou.qingshan"Ponnusamy, Saminathan"https://zbmath.org/authors/?q=ai:ponnusamy.saminathanSummary: This paper focuses on the geometric properties of quasihyperbolic geodesics in centered John metric spaces. Our contribution is to provide a necessary and sufficient condition in centered John spaces such that quasihyperbolic geodesics terminating at the center to be cone arcs. As an application, we prove that for a centered John space which carries a Gromov hyperbolic quasihyperbolization, every quasihyperbolic geodesic terminating at the center is a cone arc.Laplacian growth on branched Riemann surfaceshttps://zbmath.org/1526.300322024-02-15T19:53:11.284213Z"Gustafsson, Björn"https://zbmath.org/authors/?q=ai:gustafsson.bjorn"Lin, Yu-Lin"https://zbmath.org/authors/?q=ai:lin.yu-linThis interesting book is devoted to the Laplacian growth on Riemann surfaces. The main topics of this book are the Laplacian growth of simply connected domains in the complex plane and recent extensions obtained by its authors to the branched Riemann surfaces. There are several applications of these results in Physics, such as those related to the string equation and Hamiltonian systems.
From the classical point of view, the Laplacian growth or the Hele-Shaw problem describes the evolution (the expanding or shrinking) of a domain in the complex plane, whose boundary is evolving in the normal direction with a velocity proportional to the harmonic measure of the boundary.
The book is dedicated to the memory of Makoto Sakai, who made valuable and deep contributions to the study of extremal problems for analytic functions on Riemann surfaces, the theory of quadrature domains, and other areas of the potential theory.
The book is divided into ten chapters followed by a glossary, a list of references and an index.
In the first chapter the authors introduce the Laplacian growth and make an overview of the main topics of the book. The main subject is related to the loss of univalence of the conformal map from the unit disk onto a non-star-shaped domain, and to the construction of branched Riemann surfaces.
The second chapter is devoted to the Polubarinova-Galin and Löwner-Kufarev equations. It starts with basic results in the univalent case. The dynamics and subordination are also discussed. The last part of this chapter is devoted to the connection between the Polubarinova-Galin and Löwner-Kufarev equations in the non-univalent case. The Polubarinova-Galin equation is given by
\[
\mathrm{Re }\, \left[\dot{f}(\xi ,t)\overline{\xi f(\xi,t)}\right]=q(t),\ \ \xi \in \partial \mathbb D,
\]
where \(f(\cdot, t)\) is a conformal map of the unit disk \(\mathbb D\), such that \(f(0,t)=0\) and \(f'(0,t)>0\). In addition, \(q(t)\) is a given real-valued function representing the strength of the source/sink in the classical Hele-Shaw problem. The Löwner-Kufarev equation has the form
\[
\dot{f}(\xi ,t)=\xi f'(\xi ,t)P(\xi,t),\ \ \xi \in \mathbb{D},
\]
where
\[
P(\xi,t)=\frac{1}{2\pi i}\int _{\partial \mathbb D}\frac{q(t)}{|f'(z,t)|^2}\frac{z+\xi }{z-\xi }\frac{dz}{z},\ \ \xi \in \mathbb D,
\]
and \(f\) has a similar meaning as above.
In the third chapter the authors introduce and study weak solutions of the Polubarinova-Galin equation and their properties in terms of partial balayage related to quadrature formulas for subharmonic functions. The inverse balayage and results related to more general Laplacian evolutions are also presented.
The next chapter is concerned with weak and strong solutions on Riemann surfaces. In this chapter the authors extend the notions of Laplacian growth and partial balayage to Riemann surfaces. Useful examples are also presented.
Chapter five is devoted to global simply connected weak solutions. Theorem 5.1, which is the fundamental result of this chapter, asserts that starting with a simply connected domain in the complex plane with an analytic boundary, the corresponding Laplacian growth can be continued at any time as an evolution family of simply connected domains on some branched Riemann surface.
Chapter six studies the general structure of solutions of the Polubarinova-Galin equation and of the Löwner-Kufarev equation, when the derivative of the conformal map \(f\) is a rational function. The direct approach and the approach based on quadrature Riemann surfaces are used in the analysis developed in this chapter.
Chapter seven presents various interesting examples of Laplacian evolutions of a cardioid. There are studied both cases of the univalent and nonunivalent evolutions.
Chapter eight is concerned with the analysis of the Polubarinova-Galin equation and Laplacian growth in the context of the string equation. To this end, the string equation for univalent conformal maps is investigated.
Chapter nine deals with the Laplacian evolutions within a Hamiltonian framework, especially with the dependence of the conformal map on harmonic moments and other parameters in Hamiltonian descriptions.
Chapter ten presents a deep analysis of the Polubarinova-Galin equation in the context of the string equation for some rational functions.
This book is a valuable contribution to the modern theory of Laplacian growth. It contains many useful and interesting results, together with a rigorous analysis of all treated problems.
Reviewer: Mirela Kohr (Cluj-Napoca)On properties of zero sets of divisors in weighted spaces of entire functionshttps://zbmath.org/1526.300332024-02-15T19:53:11.284213Z"Abuzyarova, N. F."https://zbmath.org/authors/?q=ai:abuzyarova.natalya-fairbakhovna"Fazullin, Z. Yu."https://zbmath.org/authors/?q=ai:fazullin.ziganur-yusupovichSummary: In this paper we consider weighted spaces of entire functions which are Fourier-Laplace transform images of spaces of \(\Omega \)-ultradistributions. We study divisors of these spaces having only real zeros. Two properites of such zero sets are obtained. We illustrate the results by a series of examples.Geometric Julia-Wolff theorems for weak contractionshttps://zbmath.org/1526.300562024-02-15T19:53:11.284213Z"Beardon, A. F."https://zbmath.org/authors/?q=ai:beardon.alan-f"Minda, D."https://zbmath.org/authors/?q=ai:minda.davidSummary: In this paper we review the familiar collection of results that concern holomorphic maps of a disc or half-plane into itself that are due to Schwarz, Pick, Julia, Denjoy and Wolff. We give a coherent geometric treatment of these results entirely in terms of the ideas of geodesics, horocycles and G-spaces as introduced by Busemann. In particular, we show that the results of Wolff and Julia hold for all weak contractions of the hyperbolic metric (whether holomorphic or not); holomorphicity plays no role in the arguments. These results apply to holomorphic maps because the Schwarz-Pick lemma implies that holomorphic maps are weak contractions. An important ingredient in the proofs are several projections of the hyperbolic plane onto a geodesic which are weak contractions relative to the hyperbolic distance.Estimation of bounds for the zeros of polynomials and regular functions with quaternionic coefficientshttps://zbmath.org/1526.300642024-02-15T19:53:11.284213Z"Mir, Abdullah"https://zbmath.org/authors/?q=ai:mir.abdullahSummary: The aim of this work is to consider bounds for the zeros of quaternionic polynomials and regular functions with coefficients located on only one side of the quaternionic variable. We will introduce several results which put conditions on the coefficients of the polynomials and regular functions of a quaternionic variable to produce regions containing all their zeros. In particular, we extend the well known Eneström-Kakeya theorem and its various generalizations to the quaternionic setting by using the extended Schwarz's lemma and the zero sets of a regular product. The obtained results for this subclass of regular functions extend some important results for polynomials in the quaternionic variable to the case of power series.F. Wiener's trick and an extremal problem for \(H^p\)https://zbmath.org/1526.300682024-02-15T19:53:11.284213Z"Brevig, Ole Fredrik"https://zbmath.org/authors/?q=ai:brevig.ole-fredrik"Grepstad, Sigrid"https://zbmath.org/authors/?q=ai:grepstad.sigrid"Instanes, Sarah May"https://zbmath.org/authors/?q=ai:instanes.sarah-maySummary: For \(0<p \le \infty\), let \(H^p\) denote the classical Hardy space of the unit disc. We consider the extremal problem of maximizing the modulus of the \(k\)th Taylor coefficient of a function \(f \in H^p\) which satisfies \(\Vert f\Vert_{H^p}\le 1\) and \(f(0)=t\) for some \(0 \le t \le 1\). In particular, we provide a complete solution to this problem for \(k=1\) and \(0<p<1\). We also study F. Wiener's trick, which plays a crucial role in various coefficient-related extremal problems for Hardy spaces.The Heinz type inequality, Bloch type theorem and Lipschitz characteristic of polyharmonic mappingshttps://zbmath.org/1526.351412024-02-15T19:53:11.284213Z"Chen, Shaolin"https://zbmath.org/authors/?q=ai:chen.shaolin|chen.shaolin.1Summary: Suppose that \(f\) satisfies the following: (1) the polyharmonic equation \(\varDelta^m f = \varDelta (\varDelta^{m - 1} f) = \varphi_m\) \((\varphi_m \in \mathcal{C} (\overline{\mathbb{B}^n}, \mathbb{R}^n))\), (2) the boundary conditions \(\varDelta^0 f = \varphi_0, \varDelta^1 f = \varphi_1, \dots, \varDelta^{m - 1} f = \varphi_{m - 1}\) on \(\mathbb{S}^{n - 1}\) (\(\varphi_j \in \mathcal{C} (\mathbb{S}^{n - 1}, \mathbb{R}^n)\) for \(j \in \{0, 1, \dots, m - 1\}\) and \(\mathbb{S}^{n - 1}\) denotes the boundary of the unit ball \(\mathbb{B}^n\)), and (3) \(f (0) = 0\), where \(n \geq 3\) and \(m \geq 1\) are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mappings satisfying the polyharmonic equation with the above Dirichlet boundary value conditions. Furthermore, we establish a Bloch type theorem of mappings satisfying the above polyharmonic equation, which gives an answer to an open problem in [\textit{S. Chen} and \textit{S. Ponnusamy}, Indag. Math., New Ser. 30, No. 6, 1087--1098 (2019; Zbl 1443.31002)]. Additionally, we show that if \(f\) is a \(K\)-quasiconformal self-mapping of \(\mathbb{B}^n\) satisfying the above polyharmonic equation, then \(f\) is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as \(K \to 1^+\) and \(\|\varphi_j\|_\infty \to 0^+\) for \(j \in \{1, \dots, m\}\).Non-invertible dynamical systems. Volume 3. Analytic endomorphisms of the Riemann spherehttps://zbmath.org/1526.370032024-02-15T19:53:11.284213Z"Urbański, Mariusz"https://zbmath.org/authors/?q=ai:urbanski.mariusz"Roy, Mario"https://zbmath.org/authors/?q=ai:roy.mario"Munday, Sara"https://zbmath.org/authors/?q=ai:munday.saraPublisher's description: This Volume 3, is part of a series entitled Non-Invertible Dynamical Systems and presents topological dynamics, thermodynamic formalism, and fractal geometry of rational functions of the Riemann sphere. Volume 1 covers topological pressure, entropy, variational principle, equilibrium states and abstract ergodic theory. Volume 2 presents distance expanding maps, Lasota-Yorke maps and fractal geometry with applications to conformal IFSs.
This Volume 3, is part of a series entitled Non-Invertible Dynamical Systems and presents topological dynamics, thermodynamic formalism, and fractal geometry of rational functions of the Riemann sphere.
See the reviews of Volume 1 and 2 in [Zbl 1493.37003; Zbl 1505.37003].Accessibility and porosity of harmonic measure at bifurcation locushttps://zbmath.org/1526.370542024-02-15T19:53:11.284213Z"Graczyk, Jacek"https://zbmath.org/authors/?q=ai:graczyk.jacek"Świątek, Grzegorz"https://zbmath.org/authors/?q=ai:swiatek.grzegorzSummary: We study hyperbolic geodesics running from \(\infty\) to a generic point, by the harmonic measure with the pole at \(\infty \), on the boundary of the connectedness locus \(\mathcal{M}_d\) for unicritical polynomials \(f_c(z)=z^d+c\). It is known that a generic parameter \(c\in \partial\mathcal{M}_d\) is not accessible within a John angle and \(\partial\mathcal{M}_d\) spirals round them infinitely many times in both directions. We prove that almost every point from \(\partial\mathcal{M}_d\) is asymptotically accessible by a flat angle with apperture decreasing slower than \((\log \circ \dots \circ \log\operatorname{dist}(c,\partial\mathcal{M}_d))^{-1}\) for any iterate of \(\log \). This is a consequence of an iterated large deviation estimate for exponential distribution. Additionally, for an arbitrary \(\beta >0\), the bifurcation locus is not \(\beta \)-porous on a set of scales of positive density along almost every external ray with respect to the harmonic measure.A counterexample to the CFT convexity conjecturehttps://zbmath.org/1526.810362024-02-15T19:53:11.284213Z"Sharon, Adar"https://zbmath.org/authors/?q=ai:sharon.adar"Watanabe, Masataka"https://zbmath.org/authors/?q=ai:watanabe.masatakaSummary: Motivated by the weak gravity conjecture, [\textit{O. Aharony} and \textit{E. Palti}, Phys. Rev. D (3) 104, No. 12, Article ID 126005, 14 p. (2021; \url{doi:10.1103/PhysRevD.104.126005})] conjectured that in any CFT, the minimal operator dimension at fixed charge is a convex function of the charge. In this letter we construct a counterexample to this convexity conjecture, which is a clockwork-like model with some modifications to make it a weakly-coupled CFT. We also discuss further possible applications of this model and some modified versions of the conjecture which are not ruled out by the counterexample.