Lipschitz conditions, b-arcwise connectedness and conformal mappings
type: Item
instance of: scholarly article
title: Lipschitz conditions, b-arcwise connectedness and conformal mappings
zbMATH Open document ID: 0537.30006
DOI: 10.1007/BF02786870
author: Raimo Näkki
author: Bruce Palka
published in: Journal d'Analyse Mathématique
publication date: 1983-01-01T00:00:00Z
review text: Let f be a conformal mapping of a plane domain D onto the unit disk B. The authors are concerned with establishing (uniform) Hölder continuity of f or \(f^{-1}\) on D or B respectively, with Hölder exponents depending on various geometric characteristics of D. Here the geometric notions used are that of b-arcwise connectedness and some variations thereof. A set A in \({\mathbb{C}}\) is b-arcwise connected if each pair of points z and w in A can be joined by an arc \(\gamma\) in A for which \(diam(\gamma)\leq b| z-w|.\) The authors prove, for example, that if D is b-arcwise connected then f is Hölder continuous on D with exponent \(\pi /2(\pi -\arcsin(frac{1}{2}b))\) and if \({\mathbb{C}}-D\) is b- arcwise connected then \(f^{-1}\) is Hölder continuous on B with exponent \(2/(\pi^ 2(1+b)^ 2).\) The proofs involve estimates of harmonic measure and extremal length. Local versions of the results are also established.
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 3853425
zbMATH Keywords: boundary regularity
zbMATH Keywords: Hölder continuity
zbMATH Keywords: harmonic measure
zbMATH Keywords: extremal length
reviewed by: F. D. Lesley
MaRDI profile type: Publication
cites work: Quasiconformal reflections
cites work: Q5537833
cites work: Q5652949
cites work: Q5800462
cites work: On Interior and Conformal Mappings of the Disk
cites work: Q3961730
cites work: Quasiconformal circles and Lipschitz classes
cites work: Sufficient conditions for various degrees of approximation by polynomials
cites work: On the Degree of Variation in Conformal Mapping of Variable Regions
cites work: On Differentiability at the Boundary in Conformal Mapping
cites work: Q4757590
full work available at URL: https://doi.org/10.1007/bf02786870
OpenAlex ID: W1974188208
description: scientific article; zbMATH DE number 3853425