Coefficient estimates for a subclass of meromorphic functions
type: Item
instance of: scholarly article
title: Coefficient estimates for a subclass of meromorphic functions
zbMATH Open document ID: 1064.30005
author: H. M. Hossen
published in: Southeast Asian Bulletin of Mathematics
publication date: 2004-10-28T00:00:00Z
review text: For any complex number \(b\neq 0\) and \(-1\leq B < A\leq 1\) let \(F^*(b, A, B)\) denote the class of functions \(f(z) =\frac12+\sum^\infty_{k=1} a_kz^k\) analytic in \(0 < |z| < 1\) and satisfying \[ 1-\tfrac1b\,\left(\frac{zf'(z)}{f(z)}+1\right)\prec \frac{1+Az}{1+Bz} \] (\(\prec\) means subordination in \(|z| < 1\)). Theorem: If \(f\in F^*(b, A, B)\) then \[ \sum^\infty_{k=1}[(k+1)^2(1-B^2)-2(k+1)B(A - B)re\{b\}-(A-B)^2|b|^2]|a_k|^2\leq (A - B)^2|b|^2. \] If moreover \((k + 1)^2(1 - B^2)-2(k+ 1)B(A - B)re\{b\} - (A - B)^2|b|^2 > 0\) holds for \(k = 1,2,\dots,n - 2\), then \[ |a_n|\leq\frac{(A B)|b|}{n+1}\,,\quad n=1,\dots\,. \] The equality is attained for the \(n\)-th coefficient of the function \(\frac1z(1 + Bz^{n+1})^{\frac{-(A-B)b}{B(n+1)}}\), \(B\neq 0\) or \(\frac1z\,\text{exp}\,\frac{-Ab}{n+1}\,z^{n+1}\), \(B\neq 0\).
reviewed by: Jozef Waniurski
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 2111409
zbMATH Keywords: meromorphic functions
zbMATH Keywords: subordination
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2111409