Criteria for univalent functions in the unit disk
type: Item
instance of: scholarly article
title: Criteria for univalent functions in the unit disk
zbMATH Open document ID: 1269.30016
author: Saminathan Ponnusamy
author: M. Obradović
published in: Archiv der Mathematik
publication date: 2013-03-18T00:00:00Z
review text: Let \(\mathcal A\) denote the class of all normalized analytic functions in the unit disk \(\mathbb D\) and let \(\mathcal S\) be the subset of \(\mathcal A\) consisting of all functions that are univalent in \(\mathbb D\). In the paper the authors study the class \(\mathcal U\) of all functions \(f \in \mathcal A\) such that \(|U_f(z)| < 1\) for \(z \in \mathbb D\), where \[ U_f(z) = \left(\frac{z}{f(z)}\right)^2 f'(z) - 1. \] It is known that each function in \(\mathcal U\) belongs to \(\mathcal S\) and functions in \(\mathcal S\) not necessarily belong to \(\mathcal U\). In the paper two recent results concerning the membership of a function in \(\mathcal U\) are improved. One of them is the following. Let \(g, h \in \mathcal S\) and \(F(z) = g(z)h(z)/z, \,z \in \mathcal U\). Consider the problem of finding the largest value of \(r\) such that the function \(F_r(z) = r^{-1} F(rz)\) belongs to \(\mathcal U\). In a recent paper the authors obtained \(r_0 \approx 0,30294\) and they conjectured that the best possible value of \(r\) is \(1/3\). In the present paper they get \(r \approx 0,3263\) which is closer to the conjectured \(1/3\).
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C50
zbMATH DE Number: 6146014
zbMATH Keywords: analytic function
zbMATH Keywords: univalent function
zbMATH Keywords: close-to-convex function
zbMATH Keywords: starlike function
zbMATH Keywords: Schwarz lemma
reviewed by: Agnieszka Wisniowska-Wajnryb
MaRDI profile type: Publication
full work available at URL: s00013-012-0476-x
OpenAlex ID: W2076323066
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DOI: 10.1007/S00013-012-0476-X
description: scientific article; zbMATH DE number 6146014