A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions
type: Item
instance of: scholarly article
title: A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions
zbMATH Open document ID: 1169.30006
author: Toshiyuki Sugawa
author: Takao Terada
published in: Monatshefte für Mathematik
publication date: 2009-06-30T00:00:00Z
full work available at URL: http://ir.lib.hiroshima-u.ac.jp/00030650
review text: The authors use Roth's result to obtain a sharp Fekete-Szegő type inequality for \(|b_2+\mu b_1^2|\) in the class of Bloch functions \(F(z)=b_1z+b_2z^2+\dots\), \(|z|<1\), with \(\|F\|=\sup_{|z|<1}(1-|z|^2)|F'(z)|\leq1\) and \(\mu\in\mathbb C\). This is applied to derive a sharp inequality for \(|a_3|\) in the subclass \(\mathcal U(\lambda)\), \(\lambda>0\), of uniformly locally univalent functions \(f(z)=z+a_2z^2+a_3z^3+\dots\), \(|z|<1\), such that \(\sup_{|z|<1}|f''(z)/f'(z)|\leq\lambda\). More generally, \(|a_3-\mu a_2^2|\), \(\mu\in\mathbb C\), is sharply estimated for \(f\in\mathcal U(\lambda)\), \(\lambda>0\).
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 5571609
zbMATH Keywords: uniformly locally univalent function
zbMATH Keywords: Bloch function
zbMATH Keywords: pre-Schwarzian derivative
zbMATH Keywords: Fekete-Szegö inequality
zbMATH Keywords: error function
reviewed by: Dmitri V. Prokhorov
MaRDI profile type: Publication
OpenAlex ID: W2033960189
cites work: Q3812518
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DOI: 10.1007/S00605-008-0548-Y
description: scientific article; zbMATH DE number 5571609