Recent zbMATH articles in MSC 30Dhttps://zbmath.org/atom/cc/30D2023-05-31T16:32:50.898670ZUnknown authorWerkzeugMeromorphic solutions of three certain types of non-linear difference equationshttps://zbmath.org/1508.300492023-05-31T16:32:50.898670Z"Chen, Min Feng"https://zbmath.org/authors/?q=ai:chen.minfeng"Huang, Zhi Bo"https://zbmath.org/authors/?q=ai:huang.zhibo"Gao, Zong Sheng"https://zbmath.org/authors/?q=ai:gao.zongshengSummary: In this paper, the representations of meromorphic solutions for three types of nonlinear difference equations of form \( f^n(z)+P_d(z, f) = u(z)e^{v(z)}\), \(f^n(z)+P_d(z, f) = p_1e^{\lambda z}+p_2e^{-\lambda z}\) and \(f^n(z)+P_d(z, f) = p_1e^{\alpha_1z}+p_2e^{\alpha_2z}\) are investigated, where \(n\geq 2\) is an integer, \( P_d(z, f)\) is a difference polynomial in \(f\) of degree \(d\leq n-1\) with small coefficients, \( u(z)\) is a non-zero polynomial, \( v(z)\) is a non-constant polynomial, \( \lambda\), \(p_j\), \(\alpha_j\) (\(j = 1, 2\)) are non-zero constants. Some examples are also presented to show our results are best in certain sense.Rigidity of Newton dynamicshttps://zbmath.org/1508.300502023-05-31T16:32:50.898670Z"Drach, Kostiantyn"https://zbmath.org/authors/?q=ai:drach.kostiantyn"Schleicher, Dierk"https://zbmath.org/authors/?q=ai:schleicher.dierkThis well-written paper contains a detailed study of the dynamics of Newton maps of polynomials (or, to use the terminology of the paper, of polynomial Newton maps) and in particular of their rigidity properties. This is particularly interesting because polynomial Newton maps are rational maps, and the rigidity theory of rational maps is in general not as well developed as the rigidity theory of polynomials.
This paper deals with two kinds of rigidity: dynamical rigidity and parametric rigidity. Roughly speaking, dynamical rigidity concerns the possibility of distinguishing different points by means of their orbits with respect to a suitable defined symbolic dynamics, while parametric rigidity concerns instead the possibility of distinguishing (up to quasiconformal conjugation) two polynomial Newton maps by means of a suitably defined combinatorical structure of their Julia and Fatou sets.
One of the main technical advancement contained in this paper consists precisely in setting up the construction of suitable symbolic dynamics and combinatorics. This is done by introducing the notion of complex box mapping, which is a far-reaching generalization of the notion of polynomial-like maps obtained by extending ideas due to Lyubich, Kozlovski, van Strien and others. A \textit{complex box mapping} is a holomorphic map \(F\colon U\to V\) between two open sets \(U\subset V\subset\widehat{\mathbb{C}}\) with the following properties:
\begin{itemize}
\item[1.] \(F\) has finitely many critical points;
\item[2.] \(V\) is the union of finitely many open Jordan disks with disjoint closures, while \(U\) is the union of finitely or infinitely many open Jordan disks;
\item[3.] every connected component \(W\) of \(V\) either is a connected component of \(U\) or \(W\cap U\) is a union of Jordan disks compactly contained in \(W\) and with pairwise disjoint closures;
\item[4.] for every connected component \(Y\) of \(U\) the image \(F(Y)\) is a connected component of \(V\) and the restriction \(F|_Y\colon Y\to F(Y)\) is a proper map.
\end{itemize}
Using the iterated inverse images of \(V\) under \(F\), the authors are able to construct a nested sequence of puzzles, and thus a symbolic dynamics associated to this sequence of puzzles, and a notion of renormalizable dynamics. A point \(z\in U\) is \textit{non-escaping} if \(F^n(z)\in U\) for all \(n\in\mathbb{N}\); the \textit{fiber} of a non-escaping point \(z\) is the set of points in \(U\) having the same symbolic dynamics as \(z\) with respect to the given sequence of puzzles. Finally, roughly speaking, a complex box mapping has a \textit{renormalizable restriction} if there exists a puzzle piece containing a critical point so that the restriction to this puzzle piece is a polynomial-like map with connected Julia set.
The authors are then able to prove the following rigidity theorem (Theorem C) for a complex box mapping \(F\). Given an arbitrary non-escaping point \(z\), at least one of the following cases occurs:
\begin{itemize}
\item[(T)] the fiber of \(z\) is trivial, that is no other point has the same symbolic dynamics as \(z\);
\item[(R)] \(z\) belongs, possibly after applying a finite iterate, to the filled Julia set of a renormalizable restriction;
\item[(CB)] the orbit of \(z\) converges to the boundary of the domain of definition of \(F\);
\item[(NE)] \(z\) eventually maps to a periodic connected component that maps surjectively onto itself under some iterate.
\end{itemize}
The authors, then, find inside any polynomial Newton map a dynamical structure well represented by a complex box mapping (with special properties allowing to exclude some of the possibilities given by Theorem C) and use this to prove the following dynamical rigidity theorem (Theorem A) for any Newton map \(N_p\) of a polynomial \(p\) of degree at least two. Given an arbitrary \(z\in\widehat{\mathbb{C}}\), at least one of the following cases occurs:
\begin{itemize}
\item[(B)] \(z\) belongs to the basin of attraction of a root of~\(p\);
\item[(T)] the fiber of \(z\) is trivial, that is no other point has the same symbolic dynamics as \(z\);
\item[(R)] \(z\) belongs, possibly after applying a finite iterate, to the filled Julia set of renormalizable dynamics (a polynomial-like restriction of \(N_p\) with connected Julia set).
\end{itemize}
This theorem has many consequences. For instance, the authors use it to prove that the boundary of the basin of a root for a polynomial Newton map is always locally connected (this was also recently proved, in a different way, by \textit{X. Wang} et al. [``Dynamics of Newton maps'', Ergodic Theory Dyn. Syst. 43, No. 3, 1035--1080 (2023; \url{doi:10.1017/etds.2021.168})]). Furthermore, they are also able to prove that a large class of polynomial Newton maps have locally connected Julia sets; this holds, for instance, for Newton maps of cubic polynomials without Siegel disks.
Finally, the authors turn to the question of parametric rigidity. To do so, they associate to each polynomial Newton map \(N_p\) a graph called \textit{Newton graph} (very much involved also in the construction of the puzzles associated to \(N_p\)), that roughly speaking represents how the basins of the roots are connected to each other. They then say that two polynomial Newton maps are \textit{combinatorially equivalent} if their Newton graphs coincide. It turns out that two polynomial Newton maps that are quasiconfomally conjugated in a neighbourhood of their Julia set are combinatorially equivalent. Conversely, the authors are able to prove the following parametric rigidity theorem (Theorem B). If two polynomial Newton maps are combinatorially equivalent then they are quasiconformally conjugated in a neighbourhood of the Julia set provided that
\begin{itemize}
\item[(1)] either they are both non-renormalizable; or,
\item[(2)] they are both renormalizable essentially in the same way (more formally: there is a bijection between their domains of renormalization that respects hybrid equivalence between the little Julia sets as well as their combinatorial position).
\end{itemize}
Reviewer: Marco Abate (Pisa)The Hausdorff dimension of directional edge escaping points sethttps://zbmath.org/1508.300512023-05-31T16:32:50.898670Z"Huang, Xiaojie"https://zbmath.org/authors/?q=ai:huang.xiaojie"Liu, Zhixiu"https://zbmath.org/authors/?q=ai:liu.zhixiu"Li, Yuntong"https://zbmath.org/authors/?q=ai:li.yuntongSummary: In this paper, we define the directional edge escaping points set of function iteration under a given plane partition and then prove that the upper bound of Hausdorff dimension of the directional edge escaping points set of \(S(z)=a e^z +b e^{-z}\), where \(a, b\in \mathbb{C}\) and \(|a|^2 +|b|^2\neq 0\), is no more than 1.A classification of postcritically finite Newton mapshttps://zbmath.org/1508.300522023-05-31T16:32:50.898670Z"Lodge, Russell"https://zbmath.org/authors/?q=ai:lodge.russell"Mikulich, Yauhen"https://zbmath.org/authors/?q=ai:mikulich.yauhen"Schleicher, Dierk"https://zbmath.org/authors/?q=ai:schleicher.dierkSummary: The dynamical classification of rational maps is a central concern of holomorphic dynamics. Much progress has been made, especially on the classification of polynomials and some approachable one-parameter families of rational maps; the goal of finding a classification of general rational maps is so far
elusive. Newton maps (rational maps that arise when applying Newton's method to a polynomial) form a most natural family to be studied from the dynamical perspective. Using Thurston's characterization and rigidity theorem, a complete combinatorial classification of postcritically finite Newton maps is given in terms
of a finite connected graph satisfying certain explicit conditions.
For the entire collection see [Zbl 1495.57001].The analytic solutions of the functional equations \(a_1f(\gamma_1 s) + a_2f(\gamma_2 s) + \cdots + a_nf(\gamma_n s) = h(s)\)https://zbmath.org/1508.300532023-05-31T16:32:50.898670Z"Shen, Zhi-Rui"https://zbmath.org/authors/?q=ai:shen.zhi-rui"Li, Hong-Xu"https://zbmath.org/authors/?q=ai:li.hongxu|li.hong-xuSummary: In this paper, a sufficient and necessary condition of the existence of analytic solutions of the functional equation \(a_1f(\gamma_1s) + a_2f(\gamma_2s) + \cdots + a_nf(\gamma_ns) = h(s)\) for \(s\in S\) is presented, where \(S\subset\mathbb{C}\) is a given domain and \(h\) is a polynomial. This result generalizes some known results. A similar theorem on the meromorphic solutions of the equation is obtained. We also describe the structure of the set of solutions of the equation. Moreover, we give a result on the \(d\)-homogeneous solution of the homogeneous equation of the above equation for the high-dimensional case.\(D\)-module approach to Liouville's theorem for difference operatorshttps://zbmath.org/1508.300542023-05-31T16:32:50.898670Z"Cheng, Kam Hang"https://zbmath.org/authors/?q=ai:cheng.kam-hang"Chiang, Yik Man"https://zbmath.org/authors/?q=ai:chiang.yik-man"Ching, Avery"https://zbmath.org/authors/?q=ai:ching.averySummary: We establish analogues of Liouville's theorem in the complex function theory, with the differential operator replaced by various difference operators. This is done generally by the extraction of (formal) Taylor coefficients using a residue map which measures the obstruction having local ``anti-derivative''. The residue map is based on a Weyl algebra or \(q\)-Weyl algebra structure satisfied by each corresponding operator. This explains the different senses of ``boundedness'' required by the respective analogues of Liouville's theorem in this article.Möbius disjointness for a class of exponential functionshttps://zbmath.org/1508.300552023-05-31T16:32:50.898670Z"Gu, Weichen"https://zbmath.org/authors/?q=ai:gu.weichen"Wei, Fei"https://zbmath.org/authors/?q=ai:wei.feiSummary: A vast class of exponential functions is shown to be deterministic. This class includes functions whose exponents are polynomial-like or `piece-wise' close to polynomials after differentiation. Many of these functions are proved to be disjoint from the Möbius function.Integral representation of one class of entire functionshttps://zbmath.org/1508.300562023-05-31T16:32:50.898670Z"Khats', R. V."https://zbmath.org/authors/?q=ai:khats.r-vSummary: In this paper, we study an integral representation of one class of entire functions. Conditions for the existence of this representation in terms of certain solutions of some differential equations are found. We obtain asymptotic estimates of entire functions from the considered class of functions. We also give examples of entire functions from this class.On relative Ritt type and relative Ritt weak type of entire functions represented by vector valued Dirichlet serieshttps://zbmath.org/1508.300572023-05-31T16:32:50.898670Z"Datta, Sanjib Kumar"https://zbmath.org/authors/?q=ai:kumar-datta.sanjib"Biswas, Tanmay"https://zbmath.org/authors/?q=ai:biswas.tanmay"Das, Pranab"https://zbmath.org/authors/?q=ai:das.pranab-k-ii.1Summary: In this paper we wish to study some growth properties of entire functions represented by at vector valued Dirichlet series on the basis of relative Ritt type and relative Ritt weak type.Random zero sets for Fock type spaceshttps://zbmath.org/1508.300582023-05-31T16:32:50.898670Z"Kononova, Anna"https://zbmath.org/authors/?q=ai:kononova.anna-aleksandrovna|kononova.anna-vSummary: Given a nondecreasing sequence \(\Lambda = \{\lambda_n > 0\}\) such that \(\lim\limits_{n\rightarrow\infty}\lambda_n = \infty\), we consider the sequence \(\mathcal{N}_\Lambda:= \left\{\lambda_n e^{i\theta_n}, n\in\mathbb{N}\right\}\), where \(\theta_n\) are independent random variables uniformly distributed on \([0, 2\pi]\). We discuss the conditions on the sequence \(\Lambda\) under which \(\mathcal{N}_\Lambda\) is a zero set (a uniqness set) of a given weighted Fock space almost surely. The critical density of the sequence \(\Lambda\) with respect to the weight is found.Lower bound on the minimum modulus of an analytic function on a circle in terms of a negative power of its norm on a larger circlehttps://zbmath.org/1508.300592023-05-31T16:32:50.898670Z"Popov, A. Yu."https://zbmath.org/authors/?q=ai:popov.aleksandr-yurevich|popov.anton-yurevich|popov.andrei-yurevichSummary: A lower bound is derived for the maximum value of the minimum modulus of an analytic function on a circle whose radius runs through an interval with a fixed ratio of endpoints.New findings on the periodicity of entire functions and their differential polynomialshttps://zbmath.org/1508.300602023-05-31T16:32:50.898670Z"Zemirni, M. A."https://zbmath.org/authors/?q=ai:zemirni.mohamed-amine"Laine, I."https://zbmath.org/authors/?q=ai:laine.ilpo"Latreuch, Z."https://zbmath.org/authors/?q=ai:latreuch.zinelaabidineSummary: We obtain some results regarding the problem of the periodicity of entire functions \(f(z)\) when differential polynomials \(P(z, f)\) with constant coefficients generated by \(f(z)\) are periodic. We provide some sufficient conditions that ensure the periodicity of \(f(z)\), and we discuss some properties of periodic functions. Our results generalize and improve some earlier ones and have an importance concerning entire solutions of differential equations of the form \(P(z, f) = h(z)\), where \(h(z)\) is a periodic function.Further investigations on a unique range set under weight 0 and 1https://zbmath.org/1508.300612023-05-31T16:32:50.898670Z"Banerjee, A."https://zbmath.org/authors/?q=ai:banerjee.avah|banerjee.ashim|banerjee.anuradha.1|banerjee.aniruddha|banerjee.aravinda|banerjee.abhirup|banerjee.apurba|banerjee.ayan.1|banerjee.arijit|banerjee.arpan|banerjee.anurag-n|banerjee.aniket|banerjee.arindam.1|banerjee.amitabh|banerjee.amar-kumar|banerjee.aritra|banerjee.avijit|banerjee.arjun|banerjee.anjishnu|banerjee.ardhendu|banerjee.asis-kumar|banerjee.asit|banerjee.avik|banerjee.ayan|banerjee.anindita|banerjee.arindam.2|banerjee.arup|banerjee.arka|banerjee.arindam|banerjee.amitayu|banerjee.amarnath|banerjee.agnid|banerjee.anirban|banerjee.abhijit.1|banerjee.astick|banerjee.abhishek|banerjee.amartya-s|banerjee.archi|banerjee.amitava|banerjee.abhijit-vinayak|banerjee.a-d|banerjee.abhishek.1|banerjee.arnab|banerjee.asa|banerjee.amit-kumar|banerjee.ansuman|banerjee.asit-k|banerjee.abhijit|banerjee.agnijo|banerjee.ashok|banerjee.anuradha|banerjee.adrish|banerjee.anindya|banerjee.arunava|banerjee.aditya|banerjee.arvind|banerjee.ashis-gopal|banerjee.arun-k|banerjee.abhimanyu|banerjee.anandam"Maity, S."https://zbmath.org/authors/?q=ai:maity.soumen|maity.sumit|maity.samir|maity.sunil-kumar|maity.sayantan|maity.soma|maity.sudipta|maity.suman|maity.santi-prasad|maity.subhayan|maity.sushobhan|maity.soumya|maity.seba|maity.saumyen|maity.somnath|maity.sukla|maity.sayani|maity.saikat-ranjanSummary: In this paper, we have found the most generalized form of the famous Frank-Reinders polynomial. With the help of this, we have investigated the unique range set of a meromorphic function under two smallest possible weights namely 0 and 1. Our results extend some existing results in the literature.On meromorphic solutions of the Fermat-type functional equations \(f(z)^3 + f(z+c)^3 = e^p\)https://zbmath.org/1508.300622023-05-31T16:32:50.898670Z"Bi, Wenqi"https://zbmath.org/authors/?q=ai:bi.wenqi"Lü, Feng"https://zbmath.org/authors/?q=ai:lu.fengSummary: In this paper, we study the existence of meromorphic solutions of hyper-order strictly less than 1 to functional equations \(f(z)^3 + f(z+c)^3 = e^P\) over the complex plane \(\mathbb{C}\), where \(P\) is a polynomial. Meanwhile, some properties of elliptic functions are given.A note on \(L\)-order and \(L\)-lower order of meromorphic functionshttps://zbmath.org/1508.300632023-05-31T16:32:50.898670Z"Dutta, Sanjib Kumar"https://zbmath.org/authors/?q=ai:dutta.sanjib-kumar"Biswas, Arkojyoti"https://zbmath.org/authors/?q=ai:biswas.arkojyotiSummary: In this note we intend to show that if a slowly changing function \(L(r)\) is differentiable then \(L\)-order and \(L\)-lower order of a meromorphic (entire) function are respectively the same as the order and lower order of that meromorphic (entire) function.Complex flows, escape to infinity and a question of Rubelhttps://zbmath.org/1508.300642023-05-31T16:32:50.898670Z"Langley, James K."https://zbmath.org/authors/?q=ai:langley.james-kFor an entire function \(f\) the author considers the holomorphic flow \(\dot{z}=f(z)\) and the anti-holomorphic flow \(\dot{z}=\overline{f}(z)\). In a previous paper he had shown that for the holomorphic flow there are infinitely many trajectories which tend to \(\infty\) in finite time. Here he shows that such trajectories are rare in a certain sense, even though there can be uncountably many.
For the anti-holomorphic flow he constructs an example for which there are no trajectories tending to \(\infty\) in finite time. On the other hand, he shows that such trajectories exist if the inverse of the function or of its antiderivative has a logarithmic singularity over \(\infty\). In particular, this is the case if the function belongs to the Eremenko-Lyubich class.
Finally, the author shows that if the inverse of a meromorphic function \(f\) has a logarithmic singularity over \(\infty\), then there exists a path \(\gamma\) tending to \(\infty\) such that
\[
\lim_{z\to\infty,z\in\gamma} \frac{\log|f^{(m)}(z)|}{\log|z|}=\infty \text{ and }\int_\gamma |f^{(m)}(z)|^{-c}|dz|<\infty
\]
for all non-negative integers \(m\) and all \(c>0\). This is related to a question of Rubel.
Reviewer: Walter Bergweiler (Kiel)Integral formulas of Carleman and Levin for meromorphic and subharmonic functionshttps://zbmath.org/1508.300652023-05-31T16:32:50.898670Z"Men'shikova, E. B."https://zbmath.org/authors/?q=ai:menshikova.e-bSummary: When studying the interrelations between the zero distributions of holomorphic and entire functions with the addition of pole distributions for meromorphic functions and the growth of these functions, the relationships between these distributions and integral or other growth characteristics are important. In a more general subharmonic framework, these are the relationships between the Riesz measure of a subharmonic function or a Riesz charge for the difference between such functions and the growth characteristics of such functions. The basis of such relationships, as a rule, is a variety of integral formulas. A factor, which frequently complicates the use of such formulas, is the presence of normal or other derivatives of studied functions in them. This paper proposes a variant of eliminating such difficulties by inversion in the plane.Hardy-type inequalities for the Jacobi weight with applicationshttps://zbmath.org/1508.300662023-05-31T16:32:50.898670Z"Nasibullin, R. G."https://zbmath.org/authors/?q=ai:nasibullin.ramil-gaisaevichSummary: We prove some new Hardy-type inequalities for the Jacobi weight function. The resulting inequalities contain additional terms with the weight functions characteristic of Poincaré-Friedrichs inequalities. One of the constants in the inequality is unimprovable. We apply the inequalities to extending the available classes of univalent analytic functions in simply-connected domains and find univalence conditions in terms of estimates for the Schwartz derivative of an analytic function on the unit disk, the exterior of the unit disk, and the right half-plane.Recent trends in formal and analytic solutions of diff. equations. Virtual conference, University of Alcalá, Alcalá de Henares, Spain, June 28 -- July 2, 2021https://zbmath.org/1508.300672023-05-31T16:32:50.898670ZPublisher's description: This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28 -- July 2, 2021, and hosted by University of Alcalá, Alcalá de Henares, Spain.
The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, \(q\)- difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions.
The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Bruno, Alexander D.}, Normal forms of a polynomial ODE, 1-6 [Zbl 07672789]
\textit{Batkhin, Alexander B.}, Computation of homological equations for Hamiltonian normal form, 7-20 [Zbl 07672790]
\textit{Ciechanowicz, Ewa}, A note on value distribution of solutions of certain second order ODEs, 21-35 [Zbl 07672791]
\textit{Filipuk, Galina; Ligȩza, Adam; Stokes, Alexander}, Relations between different Hamiltonian forms of the third Painlevé equation, 37-42 [Zbl 07672792]
\textit{Aoki, Takashi; Uchida, Shofu}, Degeneration structure of the Voros coefficients of the generalized hypergeometric differential equations with a large parameter, 43-56 [Zbl 07672793]
\textit{Oshima, Toshio}, Riemann-Liouville transform and linear differential equations on the Riemann sphere, 57-91 [Zbl 07672794]
\textit{Cafasso, Mattia; Tarricone, Sofia}, The Riemann-Hilbert approach to the generating function of the higher order Airy point processes, 93-109 [Zbl 07672795]
\textit{Chen, Yang; Filipuk, Galina; das Neves Rebocho, Maria}, A system of nonlinear difference equations for recurrence relation coefficients of a modified Jacobi weight, 111-118 [Zbl 07672796]
\textit{Sasaki, Shoko; Takagi, Shun; Takemura, Kouichi}, \(q\)-Heun equation and initial-value space of \(q\)-Painlevé equation, 119-142 [Zbl 07672797]
\textit{Ogawara, Hiroshi}, Differential transcendence of solutions for \(q\)-difference equation of Ramanujan function, 143-153 [Zbl 07672798]
\textit{Zhang, Changgui}, On the positive powers of \(q\)-analogs of Euler series, 155-165 [Zbl 07672799]
\textit{Suwińska, Maria}, Summability of formal solutions for a family of linear moment integro-differential equations, 167-192 [Zbl 07672800]
\textit{Tahara, Hidetoshi}, Uniqueness of the solution of some nonlinear singular partial differential equations of the second order, 193-205 [Zbl 07672801]
\textit{Yoshino, Masafumi}, Solution with movable singular points of some Hamiltonian system, 207-218 [Zbl 07672802]
\textit{Lastra, Alberto; Michalik, Sławomir; Suwińska, Maria}, Some notes on moment partial differential equations. Application to fractional functional equations, 219-228 [Zbl 07672803]A power of a meromorphic function sharing a set with its higher order derivativehttps://zbmath.org/1508.300682023-05-31T16:32:50.898670Z"Karmakar, Himadri"https://zbmath.org/authors/?q=ai:karmakar.himadri"Sahoo, Pulak"https://zbmath.org/authors/?q=ai:sahoo.pulakSummary: In this paper, we deduce the form of a nonconstant meromorphic function \(f\) when some power of \(f\) shares certain set counting multiplicities in the weak sense with the \(k\)-th derivative of the power. The results of this paper generalize the results due to \textit{I. Lahiri} and \textit{S. Zeng} [Afr. Mat. 27, No. 5-6, 941--947 (2016; Zbl 1359.30043)].A note on a conjecture of R. Brückhttps://zbmath.org/1508.300692023-05-31T16:32:50.898670Z"Lahiri, Indrajit"https://zbmath.org/authors/?q=ai:lahiri.indrajit"Das, Shubhashish"https://zbmath.org/authors/?q=ai:das.shubhashishLet \(f\), \(g\) and \(a\) be entire functions (\(a\) may be a constant). \(f\) and \(g\) share the function \(a\) CM (counting multiplicities) if \(f-a\) and \(g-a\) have the same set of zeros with counting multiplicities. Let \(f\) be a nonconstant entire function with
\[
\limsup_{r\to\infty}\frac{\log\log\log M(r,f)}{\log r}<\infty,\quad \liminf_{r\to\infty}\frac{\log\log\log M(r,f)}{\log r}<\frac12,
\]
where \(M(r,f)=\max_{|z|=r}|f(z)|\), and let \(a=a(z)\) be a polynomial. It is proved, if \(f\) and \(f^{(k)}\) share \(a\) CM, then \(f^{(k)}-a=c(f-a)\), where \(c\) is a nonzero constant.
This result improves a result of \textit{Z.-X. Chen} and \textit{K. H. Shon} [Taiwanese J. Math. 8, No. 2, 235--244 (2004; Zbl 1062.30032)] concerning value sharing by an entire function with its derivative.
Reviewer: Konstantin Malyutin (Kursk)A polynomial shared by certain differential polynomialshttps://zbmath.org/1508.300702023-05-31T16:32:50.898670Z"Lahiri, Indrajit"https://zbmath.org/authors/?q=ai:lahiri.indrajit"Sinha, Kalyan"https://zbmath.org/authors/?q=ai:sinha.kalyanLet \(f\) and \(g\) be two nonconstant meromorphic functions in the complex plane \(\mathbb{C}\). If \(a\in\mathbb{C}\cup\infty\), then we denote by \(\overline{E}(a;f)\) the set of zeros of \(f-a\), and by \(E(a;f)\) we denote the set of pairs \(z,\nu\) such that \(z\) is a zero of \(f-a\) with multiplicity \(\nu\) (here the poles of \(f\) are regarded as zeros of \(f-\infty\)). The functions \(f\) and \(g\) share the value \(a\) CM (counting multiplicities) if \(E(a;f)=E(a;g)\), and they share the value \(a\) IM (ignoring multiplicities) if \(\overline{E}(a;f)=\overline{E}(a;g)\). The authors study the uniqueness problem for meromorphic functions sharing a differential polynomial.
Reviewer: Konstantin Malyutin (Kursk)Analytic functions of infinite order in half-planehttps://zbmath.org/1508.300712023-05-31T16:32:50.898670Z"Malyutin, K. G."https://zbmath.org/authors/?q=ai:malyutin.konstantin-gennadevich"Kabanko, M. V."https://zbmath.org/authors/?q=ai:kabanko.mikhail-vladimirovich"Shevtsova, T. V."https://zbmath.org/authors/?q=ai:shevtsova.tatyana-vasilevna\textit{J. Miles} [Pac. J. Math. 81, 131--157 (1979; Zbl 0371.30024)] proved that if \(f\) is an entire function of infinite order with zeros restricted to a finite number of rays, then its lower order equals infinity. In this paper, a similar result in the space of functions analytic in the upper half-plane is proved. The analytic function \(f\) in \(\mathbb{C}_+=\{z:\Im z>0\}\) is called proper analytic if \(\limsup_{z\to t}\ln|f(z)|\leq 0\) for all real numbers \(t\in\mathbb{R}\). The space of the proper analytic functions is denoted by \(JA\). The full measure of a function \(f\in JA\) is a positive measure, which justifies the term ``proper analytic function''. The order of a function \(f\in JA\) is defined as \(\displaystyle\rho=\limsup\limits_{r\to\infty}\frac{\ln(r m(r,f))}{\ln r}\), where \(\displaystyle m(r,f):=\frac 1r\int\limits_0^{\pi}\ln^+|f(re^{i\varphi})|\sin\varphi\,d\varphi\). If this limit equals infinity, then the order of the function equals infinity. In this case, the function \(f(z)\) is called functions of infinite order. Otherwise, the function \(f(z)\) is called functions of finite order. Accordingly, the lower order of a function \(f\in JA\) is defined as \(\displaystyle\underline{\rho}=\liminf\limits_{r\to\infty}\frac{\ln(r m(r,f))}{\ln r}\).
The main result is the following theorem.
Theorem. Suppose \(f\) is the proper analytic function in half-plane \(\mathbb{C}_+\) of infinite order with zeros restricted to a finite number of rays \(\mathbb{L}_k\) through the origin\,\(:\) \[\displaystyle\mathbb{L}_k=\left\{z:\arg z=e^ {i\theta_k},\quad0<\theta_k<\pi,\>k\in\overline{1,N_0},\>N_0\in\mathbb{N}\right\}\,.\] Then its lower order equals infinity.
Reviewer: Konstantin Malyutin (Kursk)Non-normality, topological transitivity and expanding familieshttps://zbmath.org/1508.300722023-05-31T16:32:50.898670Z"Meyrath, Thierry"https://zbmath.org/authors/?q=ai:meyrath.thierry"Müller, Jürgen"https://zbmath.org/authors/?q=ai:muller.jurgen.1Authors' abstract: We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel's Theorem. In particular, we also obtain characterisations of non-normality in terms of such properties.
Reviewer: Shamil Makhmutov (Muscat)Analytic order-isomorphisms of countable dense subsets of the unit circlehttps://zbmath.org/1508.300772023-05-31T16:32:50.898670Z"Burke, Maxim R."https://zbmath.org/authors/?q=ai:burke.maxim-rFor functions in \(C^k(\mathbb{R})\) which commute with a translation \(\sigma(z)=z+t\) (\(t\) is a fixed positive real number), it is proved a version of the Barth-Schneider theorem on approximation by entire functions which commute with the same translation, with a requirement that the values of the entire function and its derivatives on a specified countable set belong to specified dense sets.
Theorem 3.6. Let \((A_n^i,B_n^i),\) \(i=0,\dots,k,\) \(n\in\mathbb{N},\) be pairs of countable dense subsets of \(\mathbb{R}\) invariant under \(\sigma\) such that for each fixed \(i\), the \(A^i_n\) are pairwise disjoint. Assume also that the \(B^0_n\) are pairwise disjoint. Let \(F\subseteq\mathbb{R}\) be a finite set disjoint from each \(A_i^n\). Fix \(\varepsilon>0\). Then for each \(g\in C^k(\mathbb{R})\) with \(k\geq1\) such that \(g\sigma=\sigma g,\) \(Dg>0,\) and \(g(F)\cap B_n^0=\emptyset\) for all \(n\in\mathbb{N},\) there is an entire function \(f\) such that for \(i=0,\dots,k,\) \(n\in\mathbb{N},\) and \(x\in\mathbb{R},\)
\((a)\) \(f(\mathbb{R})\subseteq\mathbb{R},\) \(f\sigma=\sigma f,\) \(Df(x)>0;\)
\((b)\) \(|(D^if)(x)-(D^ig)(x)|<\varepsilon,\) and if \(x\in F\) then \(D^if(x)=D^ig(x);\)
\((c)\) \(D^if(A^i_n)\subseteq B^i_n,\) \(f(A^0_n)=B^0_n\).
Here \(g\sigma\) denotes the composition \(g\circ\sigma\).
Using this theorem, it is shown that if \(A\) and \(B\) are countable dense subsets of the unit circle \(T\subseteq\mathbb{C}\) with \(1\not\in A\), \(1\not\in B\), then there is an analytic function \(h:\mathbb{C}\setminus\{0\}\rightarrow\mathbb{C}\) that restricts to an order isomorphism of the arc \(T\setminus\{1\}\) onto itself and satisfies \(h(A)=B\) and \(h'(z)\neq0\) when \(z\in T\). This answers a some question (private communication) of P. M. Gauthier.
Reviewer: Konstantin Malyutin (Kursk)Universal radial limits of meromorphic functions in the unit diskhttps://zbmath.org/1508.301072023-05-31T16:32:50.898670Z"Meyrath, Thierry"https://zbmath.org/authors/?q=ai:meyrath.thierryLet \(M(\mathbb{D})\) denote the space of meromorphic functions in the unit disk \(\mathbb{D}\), including the constant function \(\infty\). Then \(M(\mathbb{D})\), equipped with the topology of locally uniform convergence with respect to the chordal metric \(\chi\) on the Riemann sphere \(\mathbb{C}_\infty\), becomes a completely metrizable space. Moreover, let \(C(\mathbb{T})\) be the space of \(\mathbb{C}_\infty\)-valued continuous functions on the unit circle \(\mathbb{T}\), equipped with the uniform metric. Complementing several known results on universal radial limit behaviour of holomorphic functions, the following theorem in the setting of spherical universality is proved: There is a dense \(G_\delta\)-set \(\mathscr{U}\) of functions \(f\) in \(M(\mathbb{D})\) having the property that for each function \(\varphi \in C(\mathbb{T})\) and for each compact set \(L\subset \mathbb{D}\) there exists an increasing sequence \((\rho_\ell)\) in \([0,1)\) such that
\[
\max_{(\zeta, z) \in \mathbb{T}\times L} \chi(f(\rho_\ell(\zeta-z)+z), \varphi(\zeta)) \to 0 \quad \mbox{ for}\quad \ell \to \infty.
\]
In addition, it is shown that any function \(f \in \mathscr{U}\) has maximal cluster set \(\mathbb{C}_\infty\) along each curve \(\gamma:[0,1) \to \mathbb{D}\) with \(\limsup_{r \to 1^-} |\gamma(r)|=1\), and that \(f\) cannot have deficient values. The proof of the main theorem hinges on a result due to Gauthier, Roth and Walsh concerning the equivalence of uniform spherical approximation and uniform norm approximation by rational functions on compact plane sets and on Mergelyan's theorem.
Reviewer: Jürgen Müller (Trier)Common universal meromorphic functions for translation and dilation mappingshttps://zbmath.org/1508.301082023-05-31T16:32:50.898670Z"Meyrath, Thierry"https://zbmath.org/authors/?q=ai:meyrath.thierryThe author proves two interesting results in the realm of universal meromorphic functions. To state these, let $M(U)$ be the space of meromorphic functions on the planar domain $U$ endowed with the topology of locally uniform convergence with respect to the chordal metric, and let $\mathbb C^*:=\mathbb C\setminus\{0\}$. For $\lambda\in \mathbb C^*$, let $T_\lambda: M(\mathbb C)\to M(\mathbb C)$ with $T_\lambda f(z):=f(z+\lambda)$ be the translation operator and $D_\lambda:M(\mathbb C^*)\to M(\mathbb C^*)$ with $D_\lambda f(z):=f(\lambda z)$ the dilation operator. If $U=\mathbb C$ or $\mathbb C^*$, a function $f\in M(U)$ is called universal for these operators $T$ if the orbit $\{T^{n} (f): n\in \mathbb N\}$ is dense in $M(U)$. Here of course $T^{n}$ is the $n$-th iterate of $T$. Denoting the set of all universal functions for the operator $T$ by $\mathcal U(T)$, it is shown that the set of common universal functions
\[
\displaystyle \bigcap_{\lambda \in \mathbb C^*} \mathcal U(T_\lambda)
\]
and
\[
\displaystyle \bigcap_{\substack{\lambda \in \mathbb C^*\\ |\lambda|\not=1}} \mathcal U(D_\lambda)
\]
are dense $G_\delta$-subsets of $M(\mathbb C)$ respectively $M(\mathbb C^*)$. It is interesting to mention that in the case of holomorphic functions on $\mathbb C^*$, there is no universal function for $D_\lambda$.
Reviewer: Raymond Mortini (Metz)Results of certain types of nonlinear delay differential equationhttps://zbmath.org/1508.341062023-05-31T16:32:50.898670Z"Liu, Man Li"https://zbmath.org/authors/?q=ai:liu.manli"Hu, Pei Chu"https://zbmath.org/authors/?q=ai:hu.peichu"Li, Zhi"https://zbmath.org/authors/?q=ai:li.zhi.4"Wang, Qiong Yan"https://zbmath.org/authors/?q=ai:wang.qiongyanSummary: We show that if the following delay differential equation
\[
[w(z+1) w(z) -1] [w(z) w(z-1)-1] + a(z) \frac{w'(z)}{w(z)} = \frac{\sum_{i=0}^p a_i (z) w^i}{\sum_{j=0}^q b_j (z) w^j}
\]
with rational coefficients \(a(z)\), \(a_i(z)\), \(b_j(z)\), admits a transcendental meromorphic solution \(w\) of finite many poles with hyper-order less than one, then it reduces into a more simple delay differential equation, which improves some known theorems obtained most recently by \textit{K. Liu} and \textit{C. J. Song} [Anal. Math. 45, No. 3, 569--582 (2019; Zbl 1449.30070)]. Moreover, we also study the delay differential equations of Tumura-Clunie type and obtain some quantitative properties of transcendental meromorphic solutions.Bases of complex exponentials with restricted supportshttps://zbmath.org/1508.420372023-05-31T16:32:50.898670Z"Lee, Dae Gwan"https://zbmath.org/authors/?q=ai:lee.dae-gwan"Pfander, Götz E."https://zbmath.org/authors/?q=ai:pfander.gotz-e"Walnut, David"https://zbmath.org/authors/?q=ai:walnut.david-fSummary: The complex exponentials with integer frequencies form a basis for the space of square integrable functions on the unit interval. We analyze whether the basis property is maintained if the support of the complex exponentials is restricted to possibly overlapping subsets of the unit interval. We show, for example, that if \(S_1, \ldots, S_K \subset [0, 1]\) are finite unions of intervals with rational endpoints that cover the unit interval, then there exists a partition of \(\mathbb{Z}\) into sets \(\Lambda_1, \ldots, \Lambda_K\) such that \(\bigcup_{k = 1}^K \{ e^{2 \pi i \lambda ( \cdot )} \chi_{S_k} : \lambda \in \Lambda_k \}\) is a Riesz basis for \(L^2 [0, 1]\). Here, \( \chi_S\) denotes the characteristic function of \(S\).On closed finite gap curves in spaceforms. IIhttps://zbmath.org/1508.530102023-05-31T16:32:50.898670Z"Klein, Sebastian"https://zbmath.org/authors/?q=ai:klein.sebastian"Kilian, Martin"https://zbmath.org/authors/?q=ai:kilian.martinSummary: We prove that the set of closed finite gap curves in hyperbolic 3-space \(\mathbb{H}^3\) is \(W^{2,2}\)-dense in the Sobolev space of all closed \(W^{2,2}\)-curves in \(\mathbb{H}^3\). We also show that the set of closed finite gap curves in any two-dimensional space form \(\mathbb{E}^2\) is \(W^{2,2}\)-dense in the Sobolev space of all closed \(W^{2,2}\)-curves in \(\mathbb{E}^2\).
For Part I, see [the authors, SIGMA, Symmetry Integrability Geom. Methods Appl. 16, Paper 011, 29 p. (2020; Zbl 1475.53025)].The random normal matrix model: insertion of a point chargehttps://zbmath.org/1508.820462023-05-31T16:32:50.898670Z"Ameur, Yacin"https://zbmath.org/authors/?q=ai:ameur.yacin"Kang, Nam-Gyu"https://zbmath.org/authors/?q=ai:kang.nam-gyu"Seo, Seong-Mi"https://zbmath.org/authors/?q=ai:seo.seong-miSummary: In this article, we study microscopic properties of a two-dimensional Coulomb gas ensemble near a conical singularity arising from insertion of a point charge in the bulk of the droplet. In the determinantal case, we characterize all rotationally symmetric scaling limits (``Mittag-Leffler fields'') and obtain universality of them when the underlying potential is algebraic. Applications include a central limit theorem for \(\log |p_n(\zeta )|\) where \(p_n\) is the characteristic polynomial of an \(n\):th order random normal matrix.