On an extreme point conjecture for concave functions
type: Item
instance of: scholarly article
title: On an extreme point conjecture for concave functions
zbMATH Open document ID: 1272.30027
DOI: 10.3103/S1066369X11120061
author: Saminathan Ponnusamy
author: Karl-Joachim Wirths
published in: Russian Mathematics
publication date: 2012-11-20T00:00:00Z
review text: Let \(1< A\leq 2\) be a fixed number, let \(\mathbb{D}= \{z: |z|< 1\}\) denote the open unit disk, \(\varphi(z): \mathbb{D}\to\overline{\mathbb{D}}\) be a holomorphic function, and define \[ \mathrm{CO}(A):= \Biggl\{f:{f''(z)\over f'(z)}= {A+1\over z-1}+ {(A-1)\varphi(z)\over 1+z\varphi(z)}\Biggr\}. \] Functions in \(\mathrm{CO}(A)\) map \(\mathbb{D}\) onto domains \(f(\mathbb{D})\) that have the opening angle at infinity of measure less than \(\pi A\) and such that the sets \(C\setminus f(\mathbb{D})\) are convex. It was conjectured by \textit{B. Bhowmik} and the authors [Indian J. Math. 50, No. 2, 339--349 (2008; Zbl 1161.30006)] that functions in \(\mathrm{CO}(A)\) that map \(\mathbb{D}\) onto the exterior of a convex unbounded polygon are among the extreme points of the closed convex hull of \(\mathrm{CO}(A)\). The conjecture remains open. The present paper bring a justification of its weak form.
reviewed by: Eligiusz Złotkiewicz
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C35
zbMATH DE Number: 6108758
zbMATH Keywords: concave univalent functions
zbMATH Keywords: starlike functions
zbMATH Keywords: extreme points
Wikidata QID: Q123202817
MaRDI profile type: Publication
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full work available at URL: s1066369x11120061
OpenAlex ID: W2072498326
description: scientific article; zbMATH DE number 6108758