On the local dilatation of quasisymmetric mappings and a theorem of Kurt Friedrichs
type: Item
instance of: scholarly article
title: On the local dilatation of quasisymmetric mappings and a theorem of Kurt Friedrichs
zbMATH Open document ID: 0573.30027
DOI: 10.1007/BF02790182
author: Richard Fehlmann
published in: Journal d'Analyse Mathématique
publication date: 1984-01-01T00:00:00Z
review text: Let \(\Omega\) be a bounded plane domain where \(\partial \Omega\) consists of free smooth arcs and curves except for finitely many corners (non-zero angles!). If \(z_ 0\in \Omega\) then there exists a constant \(b=b(\Omega,z_ 0)<1\) such that \[ | \iint_{\Omega}f(z)dxdy| \leq b\iint_{\Omega}| f(z)| dxdy \] holds for all \(f\in A(\Omega)\) (holomorphic and \(L^ 1\)-integrable functions in \(\Omega)\) which are normalized by \(f(z_ 0)=0.\) This inequality due to K. Friedrichs is proved in this paper by means of the theory of extremal quasiconformal mappings and some very elementary computations of local dilatations of quasisymmetric mappings. Then a more general statement is derived which says that there exists such a constant \(b<1\) if and only if \(\partial \Omega\) contains no substantial boundary point with respect to the boundary values induced by the affine mapping \(F_ K(x+iy)=Kx+iy\) (for an arbitrary \(K>1).\) Also bounds from below are given for \(b(\Omega,z_ 0)\), e.g. if \(\Omega\) is simply connected and \(\partial \Omega\) contains a corner of angle \(\gamma\), \(0<\gamma \leq 2\pi\), then \(b(\Omega,z_ 0)\geq | \sin \gamma | \gamma^{-1}\) for every \(z_ 0\in \Omega\). This is a kind of a converse estimation of an inequality of E. Reich.
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C75
Mathematics Subject Classification ID: 30D60
zbMATH DE Number: 3914671
zbMATH Keywords: dilatations of quasisymmetric mappings
zbMATH Keywords: inequality of E. Reich
MaRDI profile type: Publication
cites work: The boundary correspondence under quasiconformal mappings
cites work: Über extremale quasikonforme Abbildungen
cites work: On Certain Inequalities and Characteristic Value Problems for Analytic Functions and For Functions of Two Variables
cites work: Extremal Quasiconformal Mappings with Prescribed Boundary Values
cites work: On the existence of extremal Teichmüller mappings
cites work: Quasiconformal mappings with given boundary values
cites work: Q4093707
cites work: Q4093708
cites work: Q4079891
cites work: Q3310108
cites work: Q3945026
cites work: Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises
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description: scientific article; zbMATH DE number 3914671