Explicit quasiconformal extensions and Löwner chains
type: Item
instance of: scholarly article
title: Explicit quasiconformal extensions and Löwner chains
zbMATH Open document ID: 1183.30016
author: Ikkei Hotta
published in: Proceedings of the Japan Academy. Series A
publication date: 2009-12-22T00:00:00Z
review text: Let \(\mathcal A\) be the class of all analytic functions \(f\) on the unit disk \(\mathbb{D}\) normalized so that \(f(0)=f'(0)-1=0\), and, for \(k\in (0,1)\), let \(U(k) = \{ w \in \mathbb {C}: |w-c_k|< r_k\}\) with \(r_k=2k/(1-k^2)\) and \(c_k=(1+k^2)/(1-k^2)\,\). The author starts with the following well-known result. Theorem A. For \(f \in \mathcal{A}\) let \(h(z)\) represent one of the quantities \(z f'(z)/f(z)\), \(1+ f''(z)/f'(z)\), and \(f'(z)\,\). If \(h(z) \in U(k)\) for all \(z \in \mathbb{D}\,\), then the function \(f\) can be extended to a \(k\)-quasiconformal automorphism of the complex plane. In this connection, the papers of \textit{J. E. Brown} [Int. J. Math. Math. Sci. 7, 187--195 (1984; Zbl 0566.30011)], \textit{K. S. Padmanabhan} and \textit{S. Kumar} [J. Math. Phys. Sci. 25, No.4, 361--368 (1991; Zbl 0770.30019)], and \textit{T. Sugawa} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 53, 239--252 (1999; Zbl 1031.30011)] should be mentioned. The author improves this theorem by giving an explicit construction of the quasiconformal extension. The proof makes use of Löwner chains.
reviewed by: M. Vuorinen
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 5651156
zbMATH Keywords: quasiconformal extension
zbMATH Keywords: Löwner chain
MaRDI profile type: Publication
full work available at URL: pjaa.85.108
OpenAlex ID: W2092986527
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DOI: 10.3792/PJAA.85.108
description: scientific article; zbMATH DE number 5651156