Convex hulls and extreme points of classes of multivalent functions defined by gap power series
type: Item
instance of: scholarly article
title: Convex hulls and extreme points of classes of multivalent functions defined by gap power series
zbMATH Open document ID: 0556.30010
author: D. Kharzeev
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1984-01-01T00:00:00Z
review text: The authors determine the closed convex hulls and extreme points of various classes of p-valent functions whose power series have k-fold symmetric gaps. For example if \(f(z)=z^ p+\sum^{\infty}_{m=1}a_{mk+p}z^{mk+p}\) and \(Re zf'(z)/f(z)>p\alpha\) where \(\alpha <1\) then f is p-valently starlike of order \(\alpha\). The authors prove that the extreme points of the closed convex hull of this class are given by the set of functions \(\{(z^ p/(1-xz^ k)^{2p(1-\alpha)/k}):\quad | x| =1\}.\) They also give some applications to coefficient estimates. The techniques used throughout are standard.
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C50
zbMATH DE Number: 3886218
zbMATH Keywords: closed convex hulls
zbMATH Keywords: extreme points
zbMATH Keywords: p-valent functions
zbMATH Keywords: power series
zbMATH Keywords: k- fold symmetric gaps
zbMATH Keywords: starlike of order \(\alpha \)
reviewed by: David J. Hallenbeck
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3886218