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Near-Circularity in Capacity and Maximally Convergent Polynomials Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2024-03-13 Hans-Peter Blatt
If f is a power series with radius R of convergence, \(R > 1\), it is well-known that the method of Carathéodory–Fejér constructs polynomial approximations of f on the closed unit disk which show the typical phenomenon of near-circularity on the unit circle. Let E be compact and connected and let f be holomorphic on E. If \(\left\{ p_n\right\} _{n\in \mathbb {N}}\) is a sequence of polynomials converging
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About the Cover: Complex Finite Differences of Higher Order Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2024-03-06
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On the Uniqueness of L-Functions and Meromorphic Functions Under the Aegis of two Shared Sets Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2024-01-23 Sanjay Mallick, Pratap Basak
The paper presents general criteria for the uniqueness of a non-constant meromorphic function having finitely many poles and a non-constant L-function in the Selberg class when they share two sets. Our results provide the best cardinalities ever obtained in the literature improving all the existing results Li et al. (Lith. Math. J. 58(2), 249–262 (2018)), Kundu and Banerjee (Rend. Circ. Mat. Palermo
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On the Hyperbolic Metric of Certain Domains Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2024-01-17 Aimo Hinkkanen, Matti Vuorinen
We prove that if E is a compact subset of the unit disk \({{\mathbb {D}}}\) in the complex plane, if E contains a sequence of distinct points \(a_n\not = 0\) for \(n\ge 1\) such that \(\lim _{n\rightarrow \infty } a_n=0\) and for all n we have \( |a_{n+1}| \ge |a_n|/2 \), and if \(G={{\mathbb {D}}} {\setminus } E\) is connected and \(0\in \partial G\), then there is a constant \(c>0\) such that for
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Explicit Meromorphic Solutions of a Second Order Briot–Bouquet Differential Equation Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2024-01-05
Abstract In this paper, a special second-order Briot–Bouquet differential equation is studied. We construct explicit meromorphic solutions by the Kowalevski–Gambier method and a careful discussion. How we take into account the corresponding series at zeros, as opposed to the Laurent series at poles. This method is also useful for the study of many other non-linear differential equations.
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Maximizing the Second Robin Eigenvalue of Simply Connected Curved Membranes Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-12-26 Jeffrey J. Langford, Richard S. Laugesen
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Estimates of Partial Derivatives for Harmonic Functions on the Unit Disc Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-12-22
Abstract Let \(f = P[F]\) denote the Poisson integral of F in the unit disc \({{\mathbb {D}}}\) with F absolutely continuous on the unit circle \({{\mathbb {T}}}\) and \(\dot{F}\in L^p({{\mathbb {T}}})\) , where \(\dot{F}(e^{it}) = \frac{d}{dt} F(e^{it})\) . We show that for \(p\in (1,\infty )\) , the partial derivatives \(f_z\) and \(\overline{f_{\bar{z}}}\) belong to the holomorphic Hardy space \(H^p({{\mathbb
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Almost Periodic Functions: Their Limit Sets and Various Applications Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-12-22 Lev Sakhnovich
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Best Möbius Approximations of Convex and Concave Mappings Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-12-12 Martin Chuaqui, Brad Osgood
We study the best Möbius approximations (BMA) to convex and concave conformal mappings of the disk, including the special case of mappings onto convex polygons. The crucial factor is the location of the poles of the BMAs. Finer details are possible in the case of polygons through special properties of Blaschke products and the prevertices of the mapping function.
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Julia Sets, Jordan Curves and Quasi-circles Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-11-30 Norbert Steinmetz
In this paper, the classification of rational functions whose Julia sets are Jordan arcs or curves, which started in (Carleson and Gamelin in Complex dynamics, Springer, Berlin, 1993; Steinmetz in Math Ann 307:531–541, 1997), will be completed. The method of proof is based on two quasi-conformal surgery procedures, which enables shifting the critical points in simply connected (super-)attracting and
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Lattice Paths, Vector Continued Fractions, and Resolvents of Banded Hessenberg Operators Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-11-24 A. López-García, V. A. Prokhorov
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A Difference Version of the Rubel-Yang–Mues-Steinmetz–Gundersen Theorem Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-11-24 Mingliang Fang, Hui Li, Wenqiang Shen, Xiao Yao
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The Sharp Distortion Estimate Concerning Julia’s Lemma Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-11-21 Shota Hoshinaga, Hiroshi Yanagihara
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On the Forms of Meromorphic Solutions of Some Type of Non-linear Differential Equations Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-11-11 Huifang Liu, Zhiqiang Mao
In this paper, we study meromorphic solutions f of the non-linear differential equations \(f^n+P_d(z, f)=p_1e^{\alpha _1 z}+p_2e^{\alpha _2 z}+p_3e^{\alpha _3 z}\), where \(n\ge 3\) is an integer, \(P_d(z, f)\) is a differential polynomial in f of degree d, \(p_j, \alpha _j\) \(,j=1, 2, 3\), are non-zero constants. Assuming that \(d\le n-2\) or \(d=n-1\), we give the forms of meromorphic solutions
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On the Teichmüller–Nitsche Problem of T. Iwaniec, L.V. Kovalev, and J. Onninen Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-10-17 Daoud Bshouty, Abdallah Lyzzaik
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Successive Logarithmic Coefficients of Univalent Functions Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-10-11 Adam Lecko, Dariusz Partyka
The paper deals with logarithmic coefficients of univalent functions. The sharp lower and upper estimations of \(|\gamma _2(f)|-|\gamma _1(f)|\) were obtained in the class \({\mathcal {S}}\), where \(\gamma _n(f)\) denotes the n-th logarithmic coefficient of \(f\in {\mathcal {S}}\). The result is applicable to some standard subclasses of \({\mathcal {S}}\). Relevant examples were indicated.
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Clifford-Valued Ridgelet Transform: Localization Operators and Uncertainty Principles Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-10-03 Aajaz A. Teali, Firdous A. Shah
The Clifford algebra serves as a potent generalization of both Grassmann’s exterior algebra and Hamilton’s quaternion algebra in the sense that they incorporate both the geometrical and algebraic features of Euclidean space into a single structure. The goal of this article is to introduce the concept of the Clifford-valued ridgelet transform in order to utilize the benefits of ridgelet transforms for
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Non-real Zeros of Derivatives Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-10-03 J. K. Langley
A number of results are proved concerning non-real zeros of derivatives of real meromorphic functions.
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Convergence Rates of Derivatives of a Family of Barycentric Rational Hermite Interpolants for Well-Spaced Points Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-09-25 Ke Jing
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On the Monotony of Bessel Functions of the First Kind Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-09-25 Luminiţa-Ioana Cotîrlă, Róbert Szász
Let \(J_p\) denote the Bessel function of the first kind. In Baricz and András (Complex Var. Ellipti Equ. 54(7):689–696, 2009) the authors deduced a kind of monotony for a normalized form of the Bessel function \(J_p.\) They proved using integral representations that the inequalities \(-1/4
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Complex Analytic Solutions of Certain Partial Differential Equations Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-09-12 Feng Lü, Zhenliu Yang, Xinyang Liu
In this paper, we describe entire solutions of the partial differential equation $$\begin{aligned} u^m[u_{x}^{2}+2Au_{x}u_{y}+u_{y}^{2}+C_1u_x+C_2u_y+C_3u+e^h]=e^g \end{aligned}$$ in \({\mathbb {C}}^{2}\), which is a generalization of the well-known PDE of tubular surfaces. In addition, the paper is concerned with complex analytic (i.e., entire or meromorphic) solutions of a variation of the eikonal
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Boundary Smoothness Conditions for Functions in $$R^p(X)$$ Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-09-02 Stephen Deterding
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Normal Families of Meromorphic Functions in Several Variables Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-09-02 Jianming Chang, Yan Xu, Liu Yang
We extend the well-known Gu’s normality criterion for meromorphic functions of one complex variable to the case of several complex variables. Let \(\mathcal F\) be a family of functions meromorphic on a given domain \(D\subset \mathbb C^n\), k a positive integer and \(a\in (\mathbb C\setminus \{0\})^n\) a point. If for every function \(f\in \mathcal F\), all of its zeros have multiplicity at least
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Geometric Properties of Blaschke-Like Maps on Domains with a Conic Boundary Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-09-02 Masayo Fujimura, Yasuhiro Gotoh
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Maximal-Simultaneous Approximation by Faber Series in Bergman Spaces Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-08-25 Daniyal M. Israfilov
In this work maximal-simultaneous approximation properties of the partial sums of Faber series in the Bergman space of analytic functions defined on bounded continuums of the complex plane are studied. The error of this approximation in dependence of the best approximation number and parameters of the considered canonical domains is estimated.
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Planar Equilibrium Measure Problem in the Quadratic Fields with a Point Charge Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-08-23 Sung-Soo Byun
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On the Inverse Poletsky Inequality with a Cotangent Dilatation Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-08-18 Evgeny Sevost’yanov, Valery Targonskii
This article is devoted to establishing the distortion of the modulus of families of paths in wide classes of mappings that admit branch points. In particular, for mappings that are differentiable almost everywhere and have N- and \(N^{- 1}\)-Luzin properties and are absolutely continuous on almost all paths, we obtained the inverse Poletsky inequality with the so-called cotangent dilatation. We prove
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On the Condition Number of the Newton Interpolation on the Unit Disk Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-08-18 Phan Thanh Tung, Le Ngoc Cuong, Phung Van Manh
The stability of representations of univariate Lagrange interpolation polynomials in the complex plane is measured through a condition number. We study the growth of the condition number of the Newton formula for Lagrange interpolation. We prove that the condition number of Newton’s formula at the first n points of a Leja sequence for the closed unit disk \(\overline{{\mathbb {D}}}\) is bounded by
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Weighted Contractivity of Differential Operators on Fock Spaces Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-08-18 David Kalaj
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Walsh’s Conformal Map Onto Lemniscatic Domains for Polynomial Pre-images II Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-08-07 Klaus Schiefermayr, Olivier Sète
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Canonical Embeddings of Pairs of Arcs and Extremal Problems on Ring Domains Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-07-22 Alexander Yu Solynin
A configuration consisting of two disjoint Jordan arcs in \(\overline{\mathbb {C}}\) is canonical if each of these arcs is a hyperbolic geodesic segment in the domain on \(\overline{\mathbb {C}}\) complementary to the other arc. In this paper, we first show how recent results on canonical configurations obtained by M. Bonk and A. Eremenko follow from J. Jenkins’s theorem on extremal partitioning of
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An Elementary Counterexample to a Coefficient Conjecture Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-07-11 Liulan Li, Saminathan Ponnusamy, Karl-Joachim Wirths
In this article, we consider the family of functions f meromorphic in the unit disk \({\mathbb D}=\{z:\,|z| < 1\}\) with a pole at the point \(z=p\), a Taylor expansion $$\begin{aligned} f(z)= z+\sum _{k=2}^{\infty } a_kz^k, \quad |z|
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On Geometric Properties of the Ratio of Two Hypergeometric Functions Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-07-11 Toshiyuki Sugawa, Li-Mei Wang
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The Method of Boundary Value Problems in the Study of the Basis Properties of Perturbed System of Exponents in Banach Function Spaces Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-07-10 B. T. Bilalov, S. R. Sadigova, V. G. Alili
This paper considers the method of Riemann boundary value problems of the theory of analytic functions to study basis properties of perturbed systems of exponents in rearrangement invariant Banach function spaces. This method is demonstrated by the example of a system of exponents with a linear phase, depending on a complex parameter. The study of the basis properties of this system has a deep history
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An Extremal Problem for Odd Univalent Polynomials Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-06-23 Dmitriy Dmitrishin, Daniel Gray, Alexander Stokolos, Iryna Tarasenko
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A Uniqueness Problem Concerning Entire Functions and Their Derivatives Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-06-22 Andreas Sauer, Andreas Schweizer
We determine all entire functions f such that for non-zero complex values \(a \ne b\) the implications \(f=a \Rightarrow f'=a\) and \(f'=b \Rightarrow f=b\) hold. This solves an open problem in uniqueness theory. In this context, we give a normality criterion, which might be interesting in its own right.
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Composition Operators on Sobolev Spaces and Q-Homeomorphisms Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-06-01 Alexander Menovschikov, Alexander Ukhlov
In this paper, we give connections between mappings which generate bounded composition operators on Sobolev spaces and Q-mappings. Based on this, we obtain measure distortion properties of Q-homeomorphisms.
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A Local Cauchy Integral Formula for Slice-Regular Functions Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-05-30 Alessandro Perotti
We prove a local Cauchy-type integral formula for slice-regular functions. The formula is obtained as a corollary of a general integral representation formula where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two
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On the Difference Independence of the Euler Gamma Function and the Riemann Zeta Function Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-05-03 Amina Bibi, Xiao-Min Li, Hong-Xun Yi
We prove that the Riemann zeta function \(\zeta \) and the Euler gamma function \(\Gamma \) cannot satisfy a class of non-trivial algebraic difference equations with functional coefficients that are connected to the zeros of the Riemann zeta function \(\zeta \) on the critical line \(L=\{z \in \mathbb {C}: {\text {Re}}(z) = 1/2\}\). The main result of this paper is the difference analogue of the corresponding
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Bohr Type Inequalities for the Class of Self-Analytic Maps on the Unit Disk Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-02-24 Molla Basir Ahamed, Sabir Ahammed
This article is devoted to sharp improvements of the classical Bohr inequality for the class \( \mathcal {B} \) of analytic self-maps defined on the unit disk \( \mathbb {D} \). In addition, we prove a sharp result which is an improved version of the classical Bohr inequality replacing the initial coefficients \( |a_0|, |a_1| \) and \( |a_2| \) in the majorant series by \( |f(z)|, |f^{\prime }(z)|
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One Component Bounded Functions Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-02-20 Carlo Bellavita, Artur Nicolau
Three different characterizations of one-component bounded analytic functions are provided. The first one is related to the the inner-outer factorization, the second one is in terms of the size of the reproducing kernels in the corresponding de Branges–Rovnyak spaces and the last one concerns the associated Aleksandrov–Clark measure.
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Value Cross-Sharing of Meromorphic Functions Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-02-09 Funa Wang, Kai Liu
This paper concerns value cross-sharing of meromorphic functions, which is a variation to consider the uniqueness theory of meromorphic functions as usual. For example, we consider the uniqueness problems related to \(f(z)^n\) and \(g'(z)\) share common values together with \(g(z)^n\) and \(f'(z)\) share common values, or \(f(z)^n\) and \(g(z+c)^{m}\) share common values together with \(g(z)^n\) and
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About the Cover: Non-normal Families and Attracting Fixed Points Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-02-10 Elias Wegert
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Exterior John Domains and Quasisymmetric Mappings Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-01-31 Jinsong Liu, Yi Xuan
In this paper, we focus on exterior John domains. A John domain is called an exterior John domain if it is the exterior of a compact set. We prove that a quasiconformal mapping from the exterior of the closed unit ball to the exterior of a compact set is quasisymmetric with respect to the length inner distance if and only if its image is an exterior John domain. This result extends the classical results
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On the Parameter Problem of the Schwarz–Christoffel Mapping and Moduli of Quadrilaterals Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2023-01-27 G. Giorgadze, G. Kakulashvili
We analyze the Schwarz–Christoffel mapping with emphasis on its geometric properties and apply it to give an algorithm for numerical computation of the modulus of a polygonal quadrilateral. We represent the generalized modulus of the quadrilaterals by formal power series and give an algorithm for the computation of its coefficients by a hypergeometric function in a special case. To illustrate the effectiveness
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Geometric Julia–Wolff Theorems for Weak Contractions Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-12-23 A. F. Beardon, D. Minda
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Lawrence Allen Zalcman 1943–2022 Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-10-28 Mark Agranovsky, Walter Bergweiler
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Volterra-Type Integration Operators Between Weighted Bergman Spaces and Hardy Spaces Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-10-21 Yongjiang Duan, Siyu Wang, Zipeng Wang
Given an analytic function g and a \(\mathcal {D}\) weight \(\omega \) on the unit disk \(\mathbb {D}=\{z \in \mathbb {C} : |z|<1\}\), we characterize the boundedness and compactness of the Volterra-type integration operator $$\begin{aligned} J_{g}(f)(z)=\int _{0}^{z}f(\lambda )g'(\lambda )d\lambda \end{aligned}$$ between the weighted Bergman spaces \(L_{a}^{p}(\omega )\) and the Hardy spaces \(H^{q}\)
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Paatero’s Classes V(k) as Subsets of the Hornich Space Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-09-20 Valentin V. Andreev, Miron B. Bekker, Joseph A. Cima
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 \(\mathcal H\).
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On a Functional Inequality of Alzer and Salinas Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-09-16 Włodzimierz Fechner
We deal with the functional inequality $$\begin{aligned} f(x)f(y) - f(xy) \le f (x) + f (y) - f(x+y) \end{aligned}$$ for \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\), which was introduced by Horst Alzer and Luis Salinas. We show that if f is a solution that is differentiable at 0 and \(f(0)=0\), then \(f=0\) on \({\mathbb {R}}\) or \(f(x) = x\) for all \(x \in {\mathbb {R}}\). Next, we prove that every
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F. Wiener’s Trick and an Extremal Problem for $$H^p$$ H p Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-09-12 Ole Fredrik Brevig, Sigrid Grepstad, Sarah May Instanes
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New Subclasses of Univalent Mappings in Several Complex Variables: Extension Operators and Applications Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-09-09 Eduard Ştefan Grigoriciuc
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A Jentzsch-Theorem for Kapteyn, Neumann and General Dirichlet Series Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-09-09 Folkmar Bornemann
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Rational Inner Functions and their Dirichlet Type Norms Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-08-27 Linus Bergqvist
We study membership of rational inner functions in Dirichlet-type spaces in polydiscs. In particular, we prove a theorem relating such inclusions to \(H^p\) integrability of partial derivatives of an RIF, and as a corollary we prove that all rational inner functions on \({\mathbb {D}}^n\) belong to \({\mathcal {D}}_{1/n, \ldots ,1/n}({\mathbb {D}}^n)\). Furthermore, we show that if \(1/p \in {\mathcal
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On the Universality of Sequences of Differential Operators Related to Taylor Series Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-08-22 V. Lykoura, A. Mouze, V. Vlachou
For a simply connected domain \(G\subset {\mathbb {C}}\smallsetminus \{0\}\) and for a complex number \(\alpha \), with \(|\alpha |\ge 1\), we consider the sequence of operators $$\begin{aligned} T_{\alpha ,n}(f)(z)=\sum _{j=0}^{n}\frac{f^{(j)}(z)}{j!}(\alpha z)^j, \quad f\in H(G). \end{aligned}$$ We prove that this sequence of operators is hypercyclic. This problem was first investigated by Bernal-González
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Asymptotic Values of Entire Functions of Infinite Order Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-08-18 Aimo Hinkkanen, Joseph Miles
We prove that there exists an entire function for which every complex number is an asymptotic value and whose growth is arbitrarily slow subject only to the necessary condition that the function is of infinite order.
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Convergence Rates of Exceptional Zeros of Exceptional Orthogonal Polynomials Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-08-08 Brian Simanek
We consider the zeros of exceptional orthogonal polynomials (XOP). Exceptional orthogonal polynomials were originally discovered as eigenfunctions of second order differential operators that exist outside the classical Bochner–Brenke classification due to the fact that XOP sequences omit polynomials of certain degrees. This omission causes several properties of the classical orthogonal polynomial sequences
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Walsh’s Conformal Map onto Lemniscatic Domains for Polynomial Pre-images I Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-08-02 Klaus Schiefermayr, Olivier Sète
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Short Proofs for Some Classical Interpolation Theorems and Multiple Interpolation in the Disk Algebra Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-07-25 Raymond Mortini
We present short proofs of the classical interpolation theorems for derivatives by Mittag–Leffler and Germay, respectively Katsnelson, Airapetjan, Dzhrbashyan and Vinogradov and present such a theorem for the disk algebra.
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Bergman Projections Induced by Doubling Weights on the Unit Ball of $${\mathbb {C}}^n$$ C n Comput. Methods Funct. Theory (IF 2.1) Pub Date : 2022-07-18 Juntao Du, Songxiao Li, Xiaosong Liu, Yecheng Shi
Let \(p>1\) and \(\omega ,\upsilon \in {\mathcal {D}}\). The boundedness of \(P_\omega :L^\infty ({\mathbb {B}})\rightarrow {\mathcal {B}}({\mathbb {B}})\) and \(P_\omega (P_\omega ^+):L^p({\mathbb {B}},\upsilon dV)\rightarrow L^p({\mathbb {B}},\upsilon dV)\) are investigated in this paper.