The Grunsky coefficient conditions
type: Item
instance of: scholarly article
title: The Grunsky coefficient conditions
zbMATH Open document ID: 0624.30020
DOI: 10.1007/BF00970217
published in: Siberian Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: It is known that if f belongs to the class \(\Sigma\) (k) (the subset of the class \(\Sigma\), in which every function has a k-quasiconformal extension to the whole plane) then \[ (*)\quad | \sum^{\infty}_{m,n=1}\sqrt{mn}\alpha_{mn}x_ mx_ n| \leq k\sum^{\infty}_{n=1}| x_ n|^ 2 \] for every element \(x=(x_ 1,x_ 2,...)\) of the Hilbert space \(\ell^ 2\). Here \(\{\alpha_{mn}\}\) are the Grunsky coefficients of f. But the condition (*) is not sufficient for f belonging to \(\Sigma\) (k) [\textit{R. Kühnau}, Ann. Acad. Sci. Fenn., Ser. A I 6, 125-130 (1981; Zbl 0454.30016)]. In this present paper, the author gives a characterization of such functions for which the condition (*) is not only necessary but also sufficient for them to belong to \(\Sigma\) (k). As an application of such a characterization, a new example is given in which the function f satisfies the condition (*) but can only have k'- quasiconformal extensions to the whole plane with \(k'>k\).
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C75
zbMATH DE Number: 4012978
zbMATH Keywords: k-quasiconformal extension
zbMATH Keywords: Grunsky coefficients
author: Samuel L. Krushkal
MaRDI profile type: Publication
cites work: Verzerrungssätze und Koeffizientenbedingungen von GRUNSKYschen Typ für quasikonforme Abbildungen
cites work: Q4079891
full work available at URL: https://doi.org/10.1007/bf00970217
OpenAlex ID: W2332075508
description: scientific article; zbMATH DE number 4012978