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An Entire Function Weakly Sharing a Doubleton with its Derivative Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-11-25 Indrajit Lahiri
In the paper we consider the uniqueness problem of an entire function f when it shares a doubleton with its derivative \(f^{(k)}\), \(k \ge 1\). Our result extends a result of Li and Yang (J Math Soc Japan 51(4):781–799, 1999).
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Some Refinements of the Fekete and Szegő Inequality on Bounded Starlike Circular Domains in $$\mathbb {C}^n$$ C n Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-11-17 Qinghua Xu, Weiheng Feng, Taishun Liu
In this paper, by using the formula of Faà di Bruno for the k-th derivative of a composite function, we establish a refinement of the Fekete and Szegő inequality for a class of holomorphic mappings on bounded starlike circular domains in \(\mathbb {C}^n\). The results presented here generalize some known results. Finally, a certain problem is also proposed.
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Sharp Bounds for the Weighted Hölder Mean of the Zero-Balanced Generalized Complete Elliptic Integrals Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-11-13 Tie-Hong Zhao, Zai-Yin He, Yu-Ming Chu
In the article, we present the best possible bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals of the first and second kinds, which are the generalizations of previous results for complete p-elliptic integrals.
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On Uniqueness of Meromorphic Functions Partially Sharing Values with Their q -shifts Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-11-09 Pham Duc Thoan, Luong Thi Tuyet, Noulorvang Vangty
In this work, we give some uniqueness theorems for non-constant zero-order meromorphic functions when they and their q-shifts partially share values in the extended complex plane. This is a continuation of previous works of Charak et al. (J Math Anal Appl 435(2):1241–1248, 2016) and of Lin et al. (Bull Korean Math Soc 55(2):469–478, 2018). Furthermore, we show some uniqueness results in the case multiplicities
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Conformal Invariants in Simply Connected Domains Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-11-04 Mohamed M. S. Nasser, Matti Vuorinen
This paper studies the numerical computation of several conformal invariants of simply connected domains in the complex plane including, the hyperbolic distance, the reduced modulus, the harmonic measure, and the modulus of a quadrilateral. The used method is based on the boundary integral equation with the generalized Neumann kernel. Several numerical examples are presented. The performance and accuracy
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Crouzeix’s Conjecture and Related Problems Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-10-17 Kelly Bickel, Pamela Gorkin, Anne Greenbaum, Thomas Ransford, Felix L. Schwenninger, Elias Wegert
Crouzeix’s conjecture asserts that, for any polynomial f and any square matrix A, the operator norm of f(A) satisfies the estimate $$\begin{aligned} \Vert f(A)\Vert \le 2\,\sup \{|f(z)|:\ z \in W(A)\}, \end{aligned}$$(1) where \(W(A):=\{\langle Ax,x\rangle : \Vert x\Vert =1\}\) denotes the numerical range of A. This would then also hold for all functions f which are analytic in a neighborhood of W(A)
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Bank-Laine Functions with Real Zeros Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-09-24 J. K. Langley
Suppose that E is a real entire function of finite order with zeros which are all real but neither bounded above nor bounded below, such that \(E'(z) = \pm 1\) whenever \(E(z) = 0\). Then either E has an explicit representation in terms of trigonometric functions or the zeros of E have exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of
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One More Note on Neighborhoods of Univalent Functions Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-09-20 Richard Fournier
We some open questions linked to the notion of neighbourhood of a univalent function as introduced by Stephan Ruscheweyh in 1981.
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Entire Functions with Separated Zeros and 1-Points Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-09-14 Walter Bergweiler, Alexandre Eremenko
We consider transcendental entire functions of finite order for which the zeros and 1-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that such functions do not exist at all.
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Pluripotential Theory and Convex Bodies: A Siciak–Zaharjuta Theorem Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-09-08 T. Bayraktar, S. Hussung, N. Levenberg, M. Perera
We work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body P in \(({{\mathbb {R}}}^+)^d\). We define the logarithmic indicator function on \({{\mathbb {C}}}^d\): $$\begin{aligned} H_P(z):=\sup _{ J\in P} \log |z^{ J}|:=\sup _{ J\in P} \log [|z_1|^{ j_1}\cdots |z_d|^{ j_d}] \end{aligned}$$ and an associated class of plurisubharmonic (psh) functions:
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The Best Hölder-Lipschitz Constant Associated with the Function $${\varvec{(}1-z)^\alpha }$$ ( 1 - z ) α Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-08-14 Raymond Mortini, Rudolf Rupp
We determine \(\displaystyle \sup \left\{ \frac{|(1-z)^\alpha -(1-w)^\alpha |}{|z-w|^\alpha }: |z|,|w|\le 1,z\not =w\right\} \) where \(0<\alpha \le 1\).
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The Principal Branch of the Lambert W Function Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-08-13 A. F. Beardon
The Lambert W function is the multi-valued inverse of the function \(E(z) = z \exp z\). Let \(\widetilde{W}\) be a branch of W defined and single-valued on a region \(\widetilde{D}\). We show how to use the Taylor expansion of \(\widetilde{W}\) at a given point of \(\widetilde{D}\) to obtain an infinite series representation of \(\widetilde{W}\) throughout \(\widetilde{D}\).
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Blow-Up Solutions of Liouville’s Equation and Quasi-Normality Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-08-13 Jürgen Grahl, Daniela Kraus, Oliver Roth
We prove that the family \(\mathcal {F}_C(D)\) of all meromorphic functions f on a domain \(D\subseteq \mathbb {C}\) with the property that the spherical area of the image domain f(D) is uniformly bounded by \(C \pi \) is quasi-normal of order \(\le C\). We also discuss the close relations between this result and the well-known work of Brézis and Merle on blow-up solutions of Liouville’s equation.
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Valiron’s Interpolation Formula and a Derivative Sampling Formula in the Mellin Setting Acquired via Polar-Analytic Functions Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-08-11 Carlo Bardaro, Paul L. Butzer, Ilaria Mantellini, Gerhard Schmeisser
In this paper, we first recall some recent results on polar-analytic functions. Then we establish Mellin analogues of a classical interpolation of Valiron and of a derivative sampling formula. As consequences a new differentiation formula and an identity theorem in Mellin–Bernstein spaces are obtained. The main tool in the proofs is a residue theorem for polar-analytic functions.
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Entire Solutions of Two Certain Types of Non-linear Differential-Difference Equations Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-08-08 Wei Chen, Peichu Hu, Qiong Wang
In this paper, we describe entire solutions for two certain types of non-linear differential-difference equations of the form $$\begin{aligned} f^n(z)+\omega f^{n-1}(z)f'(z)+q(z)e^{Q(z)}f(z+c)=u(z)e^{v(z)}, \end{aligned}$$ and $$\begin{aligned} f^n(z)+\omega f^{n-1}(z)f'(z)+q(z)e^{Q(z)}f(z+c)=p_1e^{\lambda z}+p_2e^{-\lambda z}, \end{aligned}$$ where q, Q, u, v are non-constant polynomials, \(c,\lambda
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Meromorphic Functions Sharing four Values with their Difference Operators Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-08-07 Xiao-Min Li, Yan Liu, Hong-Xun Yi
In 2011, Heittokangas et al. (Complex Var Ellipt Equat 56(1–4):81–92, 2011) proved that if a non-constant finite order entire function f(z) and \(f(z+\eta )\) share a, b, c IM, where \(\eta \) is a finite non-zero complex number, while a, b, c are three distinct finite complex values, then \(f(z)=f(z+\eta )\) for all \(z\in \mathbb {C}\). We prove that if a non-constant finite order entire function
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Teichmüller’s Theorem in Higher Dimensions and Its Applications Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-31 Anatoly Golberg, Toshiyuki Sugawa, Matti Vuorinen
For a given ring (domain) in \(\overline{\mathbb {R}}^n\), we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all \(n\ge 3\), the standard definition of uniformly perfect sets in terms of the Euclidean metric is equivalent to the boundedness of the moduli of the separating rings. We also
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Generalization of Pólya’s Zero Distribution Theory for Exponential Polynomials, and Sharp Results for Asymptotic Growth Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-30 Janne M. Heittokangas, Zhi-Tao Wen
An exponential polynomial of order q is an entire function of the form $$\begin{aligned} f(z)=P_1(z)e^{Q_1(z)}+\cdots +P_k(z)e^{Q_k(z)}, \end{aligned}$$ where the coefficients \(P_j(z),Q_j(z)\) are polynomials in z such that $$\begin{aligned} \max \{\deg (Q_j)\}=q. \end{aligned}$$ In 1977 Steinmetz proved that the zeros of f lying outside of finitely many logarithmic strips around so called critical
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Sharp Pointwise Estimates for Fock Spaces Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-29 Friedrich Haslinger, David Kalaj, Djordjije Vujadinović
Let \(1\le p<\infty \), and \(\alpha >0\). Let \(F_{\alpha }^{p}\) denote the Fock space. We establish some sharp pointwise estimates for the derivatives of the functions belonging to \(F_{\alpha }^{p}\). Moreover for the Hilbert case \(p=2\) we establish some more specific pointwise sharp estimates. We also consider the differential operator between \(F_{\alpha }^{p}\) and \(F_{\beta }^{p}\) for \(\beta
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Exponential Integral Representations of Theta Functions Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-29 Andrew Bakan, Håkan Hedenmalm
Let \(\varTheta _{3} (z):= \sum \nolimits _{n\in \mathbb {Z}} \exp (\mathrm {i} \pi n^2 z)\) be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane \(\mathbb {H}:=\{z\in \mathbb {C}\,|\,\,\mathrm{{Im}}\, z>0\}\), and takes positive values along \( \mathrm {i} \mathbb {R}_{>0}\), the positive imaginary axis, where \(\mathbb {R}_{>0}:= (0, +\infty )\). We define
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Numerical Conformal Mapping with Rational Functions Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-28 Lloyd N. Trefethen
New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the Dirichlet problem is solved by the “lightning Laplace solver” with poles exponentially clustered near each singularity. For polygons and circular polygons, further simplifications
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On Landau’s Type Estimates for Coefficients of Univalent Functions Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-22 I. R. Kayumov, D. M. Khammatova, K.-J. Wirths
In this article we estimate the sum of coefficients for functions with restrictions on the pre-Schwarzian derivative. We obtain an estimate, which is sharp up to a constant and find upper and lower bounds for that constant. Also we estimate the sum of the first three coefficients for all functions f, such that \(\log f'(z)\) is bounded with respect to the Bloch norm.
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A Convergent Family of Bivariate Floater-Hormann Rational Interpolants Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-18 Ke Jing, Ning Kang
It is well-known that the Floater-Hormann rational interpolants give better results than other rational interpolants, especially in convergence rates and barycentric form. In this paper, we propose and study a family of bivariate Floater-Hormann rational interpolants, which have no real poles and arbitrarily high convergence rates on any rectangular region. Moreover, these interpolants are linear on
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Normal Families of Möbius Maps Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-17 A. F. Beardon, D. Minda, I. Short
Several necessary and sufficient conditions for a family of Möbius maps to be a normal family in the extended complex plane \(\mathbb {C}_\infty \) are established. Each of these conditions involves collections of two or three points which may vary with the Möbius maps in the family, provided the points satisfy a uniform separation condition. In addition, we derive a sufficient condition for the normality
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Strongly Quasiconformal Extension of Harmonic Mappings Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-15 Shuan Tang
In terms of pre-Schwarzian and Schwarzian derivatives, we obtain some sufficient conditions to ensure that a sense-preserving locally univalent harmonic mapping f in the unit disk can be extended to a quasiconformal mapping in the complex plane so that its complex dilatation satisfies some Carleson measure conditions.
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Exercises on the Theme of Continuous Symmetrization Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-13 Alexander Yu. Solynin
In this paper, we discuss the Pólya–Szegő continuous symmetrization and its applications to isoperimetric inequalities. In particular, we survey results concerning monotonicity properties of certain characteristics, including torsional rigidity of cylindrical beams and principal frequency of a uniformly stretched elastic membrane of a drum, of triangles and other domains. Several remaining open problems
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On the Functional Inequality $$f(x)f(y)-f(xy)\le f(x)+f(y)-f(x+y)$$ f ( x ) f ( y ) - f ( x y ) ≤ f ( x ) + f ( y ) - f ( x + y ) Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-04 Horst Alzer, Luis Salinas
We prove that all solutions \(f: {\mathbb {R}} \rightarrow {\mathbb {R}}\) of the functional inequality $$\begin{aligned} (*) \quad f(x)f(y)-f(xy)\le f(x)+f(y)-f(x+y), \end{aligned}$$ which are convex or concave on \({\mathbb {R}}\) and differentiable at 0 are given by $$\begin{aligned} f(x)=x\quad \text{ and } \quad f(x)\equiv c, \quad \text{ where } \quad 0\le c\le 2. \end{aligned}$$ Moreover, we
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Bohr Type Inequalities for Functions with a Multiple Zero at the Origin Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-07-04 S. Ponnusamy, K.-J. Wirths
Recently, there has been a number of good deal of research on the Bohr’s phenomenon in various settings including a refined formulation of his classical version of the inequality. Among them, in Paulsen et al. (Proc Lond Math Soc 85(2):493–512, 2002) the authors considered cases in which the above functions have a multiple zero at the origin. In this article, we present a refined version of Bohr’s
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On a Christoffel Transformation for Matrix Measures Supported on the Unit Circle Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-06-29 H. Dueñas, E. Fuentes, L. E. Garza
Let \(\sigma \) be a Hermitian matrix measure supported on the unit circle. In this contribution, we study some algebraic and analytic properties of the orthogonal matrix polynomials associated with the Christoffel matrix transformation of \(\sigma \) defined by $$\begin{aligned} d\sigma _{c_m}(z)=W_m(z)^Hd\sigma (z)W_m(z), \end{aligned}$$ where \(W_m(z)=\prod _{j=1}^m(z\mathbf{I} -A_j)\) and \(A_j\)
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The Cauchy Integral Formula in Hermitian, Quaternionic and $$\mathfrak {osp}(4|2)$$osp(4|2) Clifford Analysis Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-06-22 F. Brackx, H. De Schepper, R. Lávička, V. Souček
As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a cornerstone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as to how hermitian Clifford analysis in euclidean space \({\mathbb {R}}^{2n}\) of even dimension
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F -Classes of Univalent Functions and F -Differential Subordinations Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-05-17 José A. Antonino, Sanford S. Miller
This article introduces the F-classes of the convex, starlike and close-to-convex classes of univalent functions. The paper also extends the theory of differential subordinations to F-differential subordinations and presents examples of these new concepts.
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Deficient Values of Solutions of Linear Differential Equations Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-05-12 Gary G. Gundersen, Janne Heittokangas, Zhi-Tao Wen
Differential equations of the form \(f'' + A(z)f' + B(z)f = 0\) (*) are considered, where A(z) and \(B(z) \not \equiv 0\) are entire functions. The Lindelöf function is used to show that for any \(\rho \in (1/2, \infty )\), there exists an equation of the form (*) which possesses a solution f with a Nevanlinna deficient value at 0 satisfying \(\rho =\rho (f)\ge \rho (A)\ge \rho (B)\), where \(\rho
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On the Moisil-Theodoresco Operator in Orthogonal Curvilinear Coordinates Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-05-11 J. Bory Reyes, M. A. Pérez-de la Rosa
The action of the Moisil-Theodoresco operator over a quaternionic valued function defined on \({\mathbb {R}}^3\) (sum of a scalar and a vector field) in Cartesian coordinates is generally well understood. However this is not the case for any orthogonal curvilinear coordinate system. This paper sheds some new light on the technical aspect of the subject. Moreover, we introduce a notion of quaternionic
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Solutions of Second Order Complex Differential Equation Having Certain Pre-given Zeros Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-05-11 Fangqin Ye
For \(0
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Essential Norm of Integral Type Operators Mapping into Certain Banach Spaces of Analytic Functions Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-04-01 Amir H. Sanatpour; Maryam M. Pirasteh; Nasrin Eghbali
We study certain integral type operators acting on a large class of Banach spaces of analytic functions on the open unit disc. The operators map into weighted Banach spaces of analytic functions, Bloch type spaces or Zygmund type spaces. Besides characterizing boundedness, we give essential norm estimates of these operators. In order to investigate these operators, we also study weighted differentiation
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Meromorphic Solutions of Certain Types of Non-linear Differential Equations Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-03-28 Huifang Liu; Zhiqiang Mao
Let \(p_1, p_2\) and \(\alpha _1, \alpha _2\) be non-zero constants, and \(P_d(z, f)\) be a differential polynomial in f of degree d. Li obtained the forms of meromorphic solutions with few poles of the non-linear differential equations \(f^n+P_d(z, f)=p_1e^{\alpha _1 z}+p_2e^{\alpha _2 z}\) provided \(\alpha _1\ne \alpha _2\) and \(d\le n-2\). In this paper, given \(d=n-1\), we find the forms of meromorphic
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Embedding of $$BMOA_{\log }$$BMOAlog into Tent Spaces and Volterra Integral Operators Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-03-13 Conghui Shen; Zengjian Lou; Songxiao Li
In this paper, we study the boundedness and compactness of the identity operator \(I:BMOA_{\log }\rightarrow \mathcal {T}^{\infty }_{\log }(\mu )\). As applications, we characterize the boundedness and compactness of the Volterra integral operators \(T_g\) and \(I_g\) on the space \(BMOA_{\log }\). The estimations for the essential norm of \(T_g\) and \(I_g\) on the space \(BMOA_{\log }\) are also
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Radii Problems for Starlike Functions and Semigroup Generators Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-03-09 Mark Elin; David Shoikhet; Nikola Tuneski
In this paper, we discover new connections between topics in classical geometric function theory and complex dynamics. In particular, we study some classes of starlike functions and their embeddings in the classes of semigroup generators. In addition, we find explicit formulas for the radii of starlikeness for those classes and uniform rates of convergence of semigroups to their Denjoy–Wolff points
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The Dual Elements of Function Sets and Fefferman–Stein Decomposition of Triebel–Lizorkin Functions via Wavelets Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-02-27 Qixiang Yang; Tao Qian
Let \(D\in \mathbb {N}\), \(q\in [2,\infty )\) and \((\mathbb {R}^D,|\cdot |,dx)\) be the Euclidean space equipped with the D-dimensional Lebesgue measure. In this article, we establish the Fefferman–Stein decomposition of Triebel–Lizorkin spaces \(\dot{F}^0_{\infty ,\,q'}(\mathbb {R}^D)\) with the help of the dual on function sets which have special topological structure. A function in Triebel–Lizorkin
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Boundary Integral Formula for Harmonic Functions on Riemann Surfaces Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-02-25 Peter L. Polyakov
We construct a boundary integral formula for harmonic functions on smoothly-bordered subdomains of Riemann surfaces embeddable into \({\mathbb {C}}{\mathbb {P}}^2\). The formula may be considered as an analogue of the Green’s formula for domains in \({\mathbb {C}}\).
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Correction to: On Balayage and B-Balayage Operators Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-02-15 Maria Nowak, Paweł Sobolewski
In the original publication, article title was incorrectly published as.
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Constructing Strebel Differentials via Belyi Maps on the Riemann Sphere Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-02-08 Jijian Song; Bin Xu
In this paper, by using Belyi maps and dessin d’enfants, we construct some concrete examples of Strebel differentials with four double poles of residues 1, 1, 1, 1 on the Riemann sphere. We also prove that they have either two double zeroes or four simple zeroes. In particular, we show that they have two double zeroes if and only if their poles are coaxial; in such cases we also obtain their explicit
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The Variance of the Number of Zeros for Complex Random Polynomials Spanned by OPUC Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-02-07 Aaron M. Yeager
Let \(\{\varphi _k\}_{k=0}^\infty \) be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure \( \mu \). We study the variance of the number of zeros of random linear combinations of the form$$\begin{aligned} P_n(z)=\sum _{k=0}^{n}\eta _k\varphi _k(z), \end{aligned}$$where \(\{\eta _k\}_{k=0}^n \) are complex-valued random variables. Under the assumption that
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Boundary Behaviour of some Conformal Invariants on Planar Domains Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-02-05 Amar Deep Sarkar; Kaushal Verma
The purpose of this note is to use the scaling principle to study the boundary behaviour of some conformal invariants on planar domains. The focus is on the Aumann–Carathéodory rigidity constant, the higher order curvatures of the Carathéodory metric and a conformal metric arising from holomorphic quadratic differentials.
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Uniqueness of Meromorphic Functions Concerning their Shifts and Derivatives Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-02-03 Xiaoguang Qi; Lianzhong Yang
This work is continuation of a recent paper (Qi et al. in Comput Methods Funct Theory 18:567–582, 2018). Here, we obtain three uniqueness results of the derivative of f(z) with its shift \(f(z+c)\), leading to a new way of studying the complex differential-difference equation \(f'(z)=f(z+c)\).
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A Note on the Value Distribution of $$f^{\prime }(z)-af^k(z)- b$$f′(z)-afk(z)-b Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-01-31 Pai Yang; Jinhua Yang; Xuecheng Pang
Let \(a(\ne 0)\), b be finite complex numbers, and \(k\ge 3\) be a positive integer. If f is a meromorphic function of order \(\rho >2\), then \(f^{\prime }-af^k-b\) has infinitely many zeros on the complex plane.
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On the Convergence of Formal Exotic Series Solutions of an ODE Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-01-31 R. R. Gontsov; I. V. Goryuchkina
We propose a sufficient condition for the convergence of a complex power type formal series of the form \(\varphi =\sum _{k=1}^{\infty }\alpha _k(x^{\mathrm{i}\gamma })\,x^k\), where \(\alpha _k\) are functions meromorphic at the origin and \(\gamma \in {{\mathbb {R}}}\setminus \{0\}\), that satisfies an analytic ordinary differential equation (ODE) of a general type. An example of such a type formal
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Sharp Power Mean Inequalities for the Generalized Elliptic Integral of the First Kind Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-01-30 Miao-Kun Wang; Zai-Yin He; Yu-Ming Chu
We establish two sharp inequalities involving the power mean and generalized elliptic integral of the first kind. As applications, the analogous inequalities concerning the complete p-elliptic integral of the first kind are also derived.
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On Overconvergence of Closed to Row Subsequences of Classical Padé Approximants Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-01-29 Ralitza Kovacheva
Let \(f(z) := \sum f_\nu z^\nu \) be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Padé approximants \(\{\pi _{n,m_n}\}\) associated with f, where \(m_n\rightarrow \infty \), \(m_n\le m_{ n+1}\le m_n+1\) and \(m_n = o(n/\log n)\), resp. \(m_n = o(n)\) as \(n\rightarrow \infty \). We extend classical results
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Characterizations of Admissible Weighted Bergman Spaces on the Unit Ball Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-01-21 Sei-Ichiro Ueki
In this paper, we will give characterizations for weighted Bergman spaces with admissible weights. The first criterion is established by the invariant gradient on the unit ball of \({\mathbb {C}}^N\). The other criterion is given by the radial derivative of holomorphic functions in the unit ball.
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Boundary Behavior of Functions Representable by Weighted Koppelman Type Integral and Related Hartogs Phenomenon Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2020-01-02 C. Tryfonos; A. Vidras
In the present paper we study the boundary behavior of a weighted Koppelman type integral with a specific choice of weight for a function \(\phi \) that is integrable on a bounded domain \(D\subset \mathbb {C}^n\) and is continuous on its \(\mathcal {C}^1\)-boundary. Applying the above results, we derive a variation of Hartogs phenomena about the holomorphicity of a function \(\phi \) which is integrable
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About the Cover: The Ruscheweyh Derivatives Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2019-11-30 Elias Wegert
Honoring Stephan Ruscheweyh’s magnificent contributions to complex analysis in general and to geometric function theory in particular, the cover of this volume shows a (modified) phase plot of a Ruscheweyh derivative of the complex tangent function. In this note we sketch some background.
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Some Theorems of Approximation Theory in Weighted Smirnov Classes with Variable Exponent Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2019-11-22 Ahmet Testici
Let \({ G\subset {\mathbb {C}} }\) be a Jordan domain with rectifiable Dini smooth boundary \(\varGamma \). In this work, we investigate approximation properties of matrix transforms constructed via Faber series in weighted Smirnov classes with variable exponent. Moreover, direct and inverse theorems of approximation theory in weighted Smirnov classes with variable exponent are proved and some results
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On the Multipliers at Fixed Points of Quadratic Self-Maps of the Projective Plane with an Invariant Line Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2019-10-30 Adolfo Guillot; Valente Ramírez
This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In
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Bohr Phenomenon for Locally Univalent Functions and Logarithmic Power Series Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2019-10-19 Bappaditya Bhowmik; Nilanjan Das
In this article we prove Bohr inequalities for sense-preserving K-quasiconformal harmonic mappings defined in the unit disk \({{\mathbb {D}}}\) and obtain the corresponding results for sense-preserving harmonic mappings. In addition, Bohr inequalities are established for uniformly locally univalent holomorphic functions, and for \(\log (f(z)/z)\) where f is univalent or inverse of a univalent function
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On the Difference of Coefficients of Bazilevič Functions Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2019-09-28 Nak Eun Cho; Young Jae Sim; Derek K. Thomas
Let f be analytic in the unit disk \({\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}\), and \({\mathcal {S}}\) be the subclass of normalized univalent functions given by \(f(z)=z+\sum _{n=2}^{\infty }a_n z^n\) for \(z\in {\mathbb {D}}\). We give bounds for \(| |a_3|-|a_2| | \) for the subclass \({\mathcal B}(\alpha ,i \beta )\) of generalized Bazilevič functions when \(\alpha \ge 0\), and \(\beta \) is
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Zero Distribution on Differential-Difference Polynomials of Exponential Type Functions Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2019-09-28 Kai Liu; Linkui Gao; Xinling Liu
Exponential type functions are important subclasses of transcendental entire functions. In this paper, we will use some results given by Steinmetz (Manuscr Math 26:155–167, 1978) to consider the zeros of difference or differential-difference polynomials of exponential polynomials. In addition, we also consider the zeros of difference polynomials of exponential type functions with infinite order.
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Loewner Chains Associated with Close to Almost Starlike Mappings of Order $$\alpha $$α Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2019-09-25 Xiaofei Zhang
In this article, we give an equivalent characterization for close to almost starlike mappings of order \(\alpha \)\((0\le \alpha <1)\) in terms of Loewner chains. Next, the growth theorem and distortion theorem along a direction are obtained for this subclass of biholomorphic mappings. In particular, the results of Pfaltzgraff and Suffridge can be obtained when \(\alpha =0\).
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Uniqueness Problem of Meromorphic Mappings Sharing Moving Hyperplanes Regardless of Multiplicity Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2019-09-25 Si Duc Quang
In this article, by giving a new method to estimate the counting functions of the auxiliary function, we prove a new uniqueness theorem for degenerate meromorphic mappings sharing moving hyperplanes regardless of multiplicity. Our result extends and improves almost all results in this topic.
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Laplace Beltrami Operator in the Baran Metric and Pluripotential Equilibrium Measure: The Ball, the Simplex, and the Sphere Comput. Methods Funct. Theory (IF 0.738) Pub Date : 2019-09-07 Federico Piazzon
The Baran metric \(\delta _E\) is a Finsler metric on the interior of \(E\subset {\mathbb {R}}^n\) arising from pluripotential theory. When E is an Euclidean ball, a simplex, or a sphere, \(\delta _E\) is Riemannian. No further examples of such property are known. We prove that in these three cases, the eigenfunctions of the Laplace Beltrami operator associated with \(\delta _E\) are the orthogonal