On spherically convex univalent functions
type: Item
instance of: scholarly article
title: On spherically convex univalent functions
zbMATH Open document ID: 0969.30007
author: D. Mejía
author: Christian Pommerenke
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: A domain \(G\) on the Riemann sphere \(\overline C\) is called spherically convex if, for any pair \(w_1,w_2\in G\), the smaller arc of the greatest circle (spherical geodesic) between \(w_1\) and \(w_2\) also lies in \(G\). A meromorphic univalent function \(f\) in the unit disk \(D\) is called spherically convex \((s\)-convex) if \(f(D)\) is a spherically convex domain in \(\overline C\). In the present paper the authors prove the following useful theorem which allows to apply many results known for convex functions to \(s\)-convex functions. Theorem: Let \(f\) be univalent in \(D\) and \(f(0)=0\). Then \(f\) is \(s\)-convex if and only if the functions \(g_w(z)= f(z)/(1+ \overline w f(z))\) are convex for every \(w\in\overline {f(D)}\). As a consequence of this result they find among others new sharp bounds for \(|f(z)|\), \(|f'(z)|\) and elegant short proofs for old results on \(s\)-convex functions \(f\).
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C75
zbMATH DE Number: 1548345
zbMATH Keywords: convex functions
reviewed by: Karl-Joachim Wirths
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030374675
OpenAlex ID: W1985964443
DOI: 10.1307/MMJ/1030374675
description: scientific article; zbMATH DE number 1548345