Extremal functions for plane quasiconformal mappings
type: Item
instance of: scholarly article
title: Extremal functions for plane quasiconformal mappings
zbMATH Open document ID: 1064.30010
author: Shigenori Kurihara
author: Shinji Yamashita
published in: Journal of Mathematics of Kyoto University
publication date: 2004-12-29T00:00:00Z
review text: Let \( F(K) \) be the family of \(K\)-quasiconformal mappings \(f\) from the extended complex plane \( \overline{C} = C \cup \{\infty \}\) onto \(\overline{C} \) such that \(f(R) = R \) and \(f(-1)=-1, f(0) = 0, f(\infty)= \infty.\) The aim of the paper are the following extremal problems for a fixed \(t \in R \) \[ \lambda(K,t) = \sup_{f \in F(K)}f(t), \quad \nu(K,t) = \inf_{f \in F(K) } f(t). \] Basing on a method introduced by \textit{O. Lehto, K. I. Virtanen} and \textit{J. Väisälä} [Ann. Acad. Sci. Fenn., Ser. A I 273, 14 p. (1959; Zbl 0090.05102)] the authors determine the expressions for \(\lambda\) and \(\nu\), which are given in terms of the modulus of the Grötzsch ring. The paper contains also asymptotic results for \(t \to \infty \) and among a number of other results a chapter where the hyperbolic distance has been considered. For the last mentioned topic a reference to \textit{A. Yu. Solynin} and \textit{M. Vuorinen} [Isr. J. Math. 124, 29--60 (2001; Zbl 1021.30047)] would be appropriate.
reviewed by: Bodo Dittmar
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C75
zbMATH DE Number: 2124029
zbMATH Keywords: quasiconformal mapping in the plane
zbMATH Keywords: extremal problem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250283741
OpenAlex ID: W1558904455
DOI: 10.1215/KJM/1250283741
description: scientific article; zbMATH DE number 2124029