A nonlinear extremal problem for Bloch functions with applications to geometric function theory
type: Item
instance of: scholarly article
title: A nonlinear extremal problem for Bloch functions with applications to geometric function theory
zbMATH Open document ID: 1041.30010
author: Oliver Roth
published in: Archiv der Mathematik
publication date: 2004-03-07T00:00:00Z
review text: Let \(D\) be a domain in the complex plane equipped with a hyperbolic density \(\lambda_D\). For fixed \(C\in \mathbb R\) the author finds an upper bound for \(\sup_{w\in D}| g''(w)-Cg'(w)^2| /\lambda_D(w)^2\), where \(g\) is an analytic Bloch function in \(D\). This bound is best possible if \(D\) is simply connected. As a corollary the author obtains a sharp upper estimate for the norm of the Schwarzian derivative of an analytic function in a simply connected domain in terms of the norm of its pre-Schwarzian. This yields a nonlinear extension of a theorem of \textit{K-J. Wirths} [Arch. Math. 30, 606--612 (1978; Zbl 0373.30016)] and also a converse of a result of \textit{M. Chuaqui} and \textit{B. Osgood} [Comment. Math. Helv. 69, No. 4, 659--668 (1994; Zbl 0826.30013)]. As another corollary, the sharp upper bound of the third Taylor coefficient in the set of analytic functions \(f: \mathbb D \to \mathbb C\) satisfying \((1-| z| ^2)| f''(z)/f'(z)| \leq 1\) and normalized by \(f(0)=f'(0)-1=0\) is obtained.
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30C55
zbMATH DE Number: 2051651
zbMATH Keywords: Bloch function
zbMATH Keywords: hyperbolic domain
zbMATH Keywords: extremal problem
reviewed by: Zengjian Lou
MaRDI profile type: Publication
full work available at URL: s00013-003-4799-5
OpenAlex ID: W1999037199
DOI: 10.1007/S00013-003-4799-5
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description: scientific article; zbMATH DE number 2051651