On the coefficients of powers of a class of Bazilevich
type: Item
instance of: scholarly article
title: On the coefficients of powers of a class of Bazilevich
zbMATH Open document ID: 0607.30013
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let S denote the class of functions \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) analytic and univalent in the open unit disk \(U=\{z:| z| <1\}\). For \(\alpha >0\) and f(z) in S, let \[ \{\frac{f(z)}{z}\}^{1/\alpha} = \sum^{\infty}_{n=0}a_ n(\alpha)z^ n\text{ and } \{\frac{k(z)}{z}\}^{1/\alpha} = \sum^{\infty}_{n=0}b_ n(\alpha)z^ n, \] where \(k(z)=z/(1-z)^ 2\) (z\(\in U)\) is the well-known Koebe function of class S. For \(\beta >0\) the class B(\(\beta)\) of Bazilevich functions of type \(\beta\) consists of functions f(z) analytic in U and satisfying \[ zf'(z)\{f(z)\}^{\beta - 1}=\{g(z)\}^{\beta}p(z), \] where g(z) is a normalized starlike function so that \(zg'(z)/g(z)=q(z)\). Here p(z) and q(z) are functions of positive real part with \(p(0)=q(0)=1\). It is known that functions in B(\(\beta)\) are univalent and that B(1) is the class of normalized close- to-convex functions defined by W. Kaplan. In this paper the Bazilevich class B(1) receives special consideration, and for f(z)\(\in B(1)\) sharp estimates for the coefficients \(a_ 2(\alpha)\) are given. Some sharp coefficient estimates for functions in B(m), m a positive integer, are also included. Particular attention is paid to cases in which the appropriate power of the Koebe function is not extremal and also to the validity of the inequality \[ | a_ n(\alpha)| \leq b_ n(\alpha). \]
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C50
zbMATH DE Number: 3981493
zbMATH Keywords: Bazilevich functions
zbMATH Keywords: functions of positive real part
zbMATH Keywords: close-to-convex functions
author: Massoud Jahangiri
reviewed by: Malcolm T. McGregor
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3981493