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Coefficients inequalities for classes of meromorphic functions

A typical problem in the theory of analytic functions is to study a functional made up of combinations of coefficients of the original function. Usually, there is a parameter over which the extremal value of the functional is needed. One of the important functionals of this type is the Fekete–Szegö functional defined on the class of analytic functions. In this paper we transfer the Fekete–Szegö problem to some classes of meromorphic functions. | Turk J Math (2018) 42: 2506 – 2512 © TÜBİTAK doi:10.3906/mat-1703-100 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Coefficients inequalities for classes of meromorphic functions Jacek DZIOK1,∗,, Maslina DARUS2 ,, Janusz SOKÓŁ3 , Faculty of Mathematics and Natural Sciences, University of Rzeszów, Rzeszów, Poland 2 Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor Darul Ehsan, Malaysia 3 Faculty of Mathematics and Natural Sciences, University of Rzeszów, Rzeszów, Poland 1 Received: 23.03.2017 • Accepted/Published Online: 15.07.2018 • Final Version: 27.09.2018 Abstract: A typical problem in the theory of analytic functions is to study a functional made up of combinations of coefficients of the original function. Usually, there is a parameter over which the extremal value of the functional is needed. One of the important functionals of this type is the Fekete–Szegö functional defined on the class of analytic functions. In this paper we transfer the Fekete–Szegö problem to some classes of meromorphic functions. Key words: Meromorphic functions, Fekete–Szegö problem, subordination, Hadamard product 1. Introduction Let A denote the class of functions that are analytic in D = {z ∈ C: |z| 0, χn − ψn > 0, p1 , q1 > 0 (n = 0, 1, .). ∗Correspondence: jdziok@ur.edu.pl 2010 AMS Mathematics Subject Classification: 30C45, 30C50, 30C55 2506 This work is licensed under a Creative Commons Attribution 4.0 International License. (2) (z ∈ D)} . DZIOK et al./Turk J Math By Σα (ϕ, φ; ψ, χ; p) we denote the class of functions f ∈ Σ such that (φ ∗ f ) (z) (χ ∗ f ) (z) ̸= 0 (z ∈ ∆) and (1 − α) ϕ∗f ψ∗f +α ≺ p, φ∗f χ∗f where ∗ denotes the Hadamard product or convolution and ≺ is the symbol of subordination. Moreover, let us define ∞ 1 ∑ n Σ∗ (p) := Σ0 (−zφ′ (z) , φ; φ, φ; p) , φ (z) = + z . z n=0 The class Σα (ϕ, φ; ψ, χ; p) is related to the classes of meromorphic Mocanu functions, meromorphic .