Inverse functions of Grötzsch's and Teichmüller's modulus functions
type: Item
instance of: scholarly article
title: Inverse functions of Grötzsch's and Teichmüller's modulus functions
zbMATH Open document ID: 1077.30007
author: Shinji Yamashita
published in: Journal of Mathematics of Kyoto University
publication date: 2005-01-03T00:00:00Z
review text: The Grötzsch ring domain is the planar doubly-connected domain \(\{z:| z| <1\}\setminus [0,r]\), \(0<r<1\). It can be mapped conformally onto the annulus \(\{z:1<| z| <e^{\mu(r)}\}\), where \(\mu\) is the Grötzsch modulus function which can be expressed in terms of elliptic integrals. Let \(\chi:(0,\infty)\to (0,1)\) be the inverse function of \(\mu\). We also consider the Teichmüller modulus function \(T(t)= 2\mu(1/\sqrt{1+t})\), \(t>0\) and its inverse \(S(x)=\chi(x/2)^{-2}-1,\;x>0\). Let also \(\sigma_n\) denote the \(n\)th iteration of the function \(\sigma(r)=2\sqrt{r}/(1+r)\), \(r>0\). The author proves error bounds for the approximation \(\chi(x)^\beta\approx \sigma_n(4e^{-2^nx})^\beta\) for \(\beta\geq -2,\,\beta\neq 0\). Let \(\lambda(K)=S(KT(1))\), \(K\geq 1\). This function appears in the theory of \(K\)-quasiconformal maps. The author proves that the above mentioned error bounds lead to the inequalities \[ 1.2425\dots<(\lambda(K)-16^{-1}e^{\pi K}+2^{-1})e^{\pi K}<1.25,\quad K\geq 1. \] The right hand inequality is sharp and improves earlier results that appear in the book ``Conformal invariants, inequalities, and quasiconformal maps'' (1997; Zbl 0885.30012) by \textit{G. D. Anderson, M. K. Vamanamurthy}, and \textit{M. K. Vuorinen}. In this book one can find more details about the Grötzsch and Teichmüller functions, their expressions in terms of elliptic functions, and their use in quasiconformal maps. Yamashita also studies the Lipschitz continuity of \(\chi\) and \(S\) and their expressions in terms of elliptic theta functions.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 2124478
zbMATH Keywords: Grötzsch modulus function
zbMATH Keywords: Teichmüller modulus function
zbMATH Keywords: elliptic theta function
zbMATH Keywords: quasiconformal mapping
zbMATH Keywords: elliptic integral
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250281735
OpenAlex ID: W1593492147
DOI: 10.1215/KJM/1250281735
description: scientific article; zbMATH DE number 2124478