Partial sums of starlike and convex functions
type: Item
instance of: scholarly article
title: Partial sums of starlike and convex functions
zbMATH Open document ID: 0894.30010
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-08-30T00:00:00Z
review text: Let \(S\) be the class of functions \(f:f(z)= z+\sum^\infty_{k=2} a_kz^k\) that are analytic in the unit disc \(E\). Let \(S^*(\alpha)\) and \(K(\alpha)\), respectively, be the subclasses of \(S\) which consist of starlike and convex functions of order \(\alpha, 0\leq\alpha <1\). A sufficient condition for \(f\) to be in \(S^*(\alpha)\) is that \(\sum^\infty_{k=2} (k-\alpha) | a_k |\leq 1-\alpha\) and to be in \(K(\alpha)\) is that \(\sum^\infty_{k=2} K(k-\alpha) | a_k|\leq 1- \alpha.\) In this paper, the ratio of a function \(f\) to its sequence of partial sums \(f_n(z)= z+ \sum^n_{k=2} a_kz^k\) is studied when the coefficients of \(f\) are sufficiently small to satisfy one of these conditions. Sharp lower bounds for \(\text{Re} \left\{{f(z) \over f_n(z)}\right\}\), \(\text{Re} \left\{{f_n(z) \over f(z)}\right\}\), \(\text{Re} \left\{{f'(z) \over f_n'(z)}\right\}\) and \(\text{Re} \left\{ {f_n'(z) \over f'(z)}\right\}\) are determined when the coefficients \(\{a_k\}\) are small.
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 1031564
zbMATH Keywords: partial sums
zbMATH Keywords: starlike
zbMATH Keywords: convex functions
author: H. Silverman
reviewed by: Khalida Inayat Noor
MaRDI profile type: Publication
OpenAlex ID: W2036871306
cites work: Convex Hulls and Extreme Points of Families of Starlike and Convex Mappings
cites work: A Note on the Partial Sums of Convex Schlicht Functions
cites work: Univalent Functions with Negative Coefficients
cites work: Q3728316
DOI: 10.1006/JMAA.1997.5361
description: scientific article; zbMATH DE number 1031564