An interpolating family of a univalent analytic function
type: Item
instance of: scholarly article
title: An interpolating family of a univalent analytic function
zbMATH Open document ID: 0659.30023
DOI: 10.1007/BF00969319
published in: Siberian Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Delta\) be the open unit disk and \(\Delta^*\) the complement of the closed unit disk in the complex plane. Let \(\Sigma\) be the class of holomorphic functions which are univalent in \(\Delta^*\) and with development \[ f(z)=z+a_ 0+a_ 1z^{-1}+...\quad. \] Let \(\Sigma_ k\) denote the subclass of \(\Sigma\) consisting of univalent functions f which admit a k'-quasiconformal extension \(\hat f\) to \(\Delta\) with k'\(\leq k.\) The formula \(f_ t(z)=tf(z/t)\), \(0\leq | t| \leq 1\), is an isotopy connecting f to the identity. The author answers positively a question of Bshouty by proving the following theorem about this isotopy. Theorem. If f is in \(\Sigma_ k\), then for any complex number t with \(| t| <1\), \(f_ t\) is in \(\Sigma_{k| t|}\).
Mathematics Subject Classification ID: 30C99
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 4077647
zbMATH Keywords: schlicht functions
zbMATH Keywords: quasiconformal extension
author: Samuel L. Krushkal
reviewed by: Frederick P. Gardiner
MaRDI profile type: Publication
cites work: Q3961718
cites work: Approximation of Infinite Dimensional Teichmuller Spaces
cites work: Curves von 2-manifolds and isotopies
cites work: Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen.
Wikidata QID: Q126027147
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description: scientific article; zbMATH DE number 4077647