Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions
type: Item
instance of: scholarly article
title: Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions
zbMATH Open document ID: 0944.30016
published in: Commentarii Mathematici Helvetici
publication date: 2000-09-18T00:00:00Z
review text: Let \(f\) be a sense-preserving quasisymmetric homeomorphism of the unit circle onto itself, \(K_1(f)\) be the maximal dilatation of the extremal extension of \(f\) to the unit disk, and \(H(f)\) be the boundary dilatation of \(f\). Define \(K_0(f)=\sup_Q\{{M(f(Q))\over M(Q)}\}\) where sup is taken over all quadrilaterals \(Q\) with the unit disk as its domain, and \(M(Q)\) is the conformal modulus of \(Q\). It is well known that \(K_0(f)\leq K_1(f)\) and \(H(f)\leq K_1(f)\). If there is a quadrilateral \(Q\) such that \(K_0(f)={M(f(Q))\over M(Q)}\), then we denote \(K_0(f)\) by \(K^q_0(f)\); if \(K_0(f)>{M(f(Q))\over M(Q)}\) for every \(Q\), then we denote \(K_0(f)\) by \(K^d_0(f)\). In this paper, the author discusses the relationship among \(K_0(f)\), \(K^q_0(f)\), \(K_0^d(f)\), \(K_1(f)\) and \(H(f)\), and obtain some interesting results such as: Theorem 1. Either \(K_0(f)= K^q_0 (f)\), or \(K^d_0(f)\leq H(f)\). Theorem 2. If \(K^q_0(f)=K_1(f)\), then \([f]\) is a Strebel point in the universal Teichmüller space. Theorem 4. If \(H(f)=1\) and \(f\neq id\), then \(K_0(f)<K_1(f)\).
Mathematics Subject Classification ID: 30C75
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 1129400
zbMATH Keywords: quasisymmetric function
zbMATH Keywords: modulus of quadrilateral
zbMATH Keywords: boundary dilatation
zbMATH Keywords: Strebel point
author: Shengjian Wu
reviewed by: Jixiu Chen
MaRDI profile type: Publication
full work available at URL: s000140050036
OpenAlex ID: W2004803067
DOI: 10.1007/S000140050036
description: scientific article; zbMATH DE number 1129400