On conformal mappings onto quasidisks
type: Item
instance of: scholarly article
title: On conformal mappings onto quasidisks
zbMATH Open document ID: 1201.30022
DOI: 10.1007/BF03321764
author: Ngin-Tee Koh
published in: Computational Methods and Function Theory
publication date: 2010-06-17T00:00:00Z
review text: The paper under review studies the growth behavior of conformal mappings of the exterior \(\Delta^*=\mathbb{C}\cup\{\infty\}\setminus\overline{\Delta}\) of the unit disc \(\Delta\subset\mathbb{C}\). It is shown that if \(g:\Delta^*\to D\) is a conformal mapping with \(D\) a quasi-disc (and hence \(g\) has a \(K\)-quasiconformal extension to \(\mathbb{C}\cup\{\infty\}\)), then there are positive constants \(C=C(K)\), \(\beta=\beta(K)\) such that whenever \(z\in\Delta^*\cap\mathbb{C}\), \[ \text{dist}(g(z),\partial D)\leq (|z|^2-1)|g^\prime(z)| \leq C\, (|z|+1)^\beta \, \text{dist}(g(z),\partial D). \] A key tool used in the proof is the well-known fact that quasiconformal mappings of \(\mathbb{C}\cup\{\infty\}\) are quasisymmetric. As an application, the paper studies the near-boundary behavior of particular quasiconformal extensions of conformal mappings on bounded quasi-discs.
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C20
zbMATH DE Number: 5723309
zbMATH Keywords: conformal mapping
zbMATH Keywords: quasiconformal mapping
zbMATH Keywords: quasi-disc
zbMATH Keywords: Douady-Earle extension
reviewed by: Nageswari Shanmugalingam
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5723309