Grunsky coefficient inequalities, Carathéodory metric and extremal quasiconformal mappings
type: Item
instance of: scholarly article
title: Grunsky coefficient inequalities, Carathéodory metric and extremal quasiconformal mappings
zbMATH Open document ID: 0697.30016
DOI: 10.1007/BF02564699
published in: Commentarii Mathematici Helvetici
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140176
review text: The following results are obtained: I. There is given the complete solution of the question of how to characterize a class of functions f, for which the sharpened Grunsky inequality \[ (*)\quad | \sum^{\infty}_{m,n=1}\sqrt{mn}\alpha_{mn}(f)x_ mx_ n| \leq k\| x\|^ 2,\quad x=(x_ 1,x_ 2,...)\in I^ 2, \] is both necessary and sufficient to have a k-quasiconformal extension onto the whole sphere \({\bar {\mathbb{C}}}\). For example, in \(\Delta^*=\{z\in {\bar {\mathbb{C}}}:\) \(| z| >1\}\) that are only the functions which are the restrictions to \(\Delta^*\) of the quasiconformal automorphisms \(w^{\mu}\) of \({\bar {\mathbb{C}}}\) with complex dilatation \(\mu\) satisfying the equality \[ \sup \{| \iint_{\Delta}\mu \phi dz\wedge d\bar z|:\quad \| \phi \|_{A_ 1}=1\}=\| \mu \|_{\infty}, \] where \(A_ 1\) is the subspace of \(L_ 1(\Delta)\) formed by holomorphic functions and, in addition, \(\phi\) are the squares of functions holomorphic in \(\Delta\), \(\Delta =\{| z| <1\}\). The proof uses the properties of the Carathéodory metric of the universal Teichmüller space T. II. If \(\phi \in A_ 1\setminus \{0\}\) has zeros of even order only, then in the holomorphic disk in T, defined by the Beltrami differential t\({\bar \phi}\)/\(| \phi |\), \(t\in \Delta\), the Carathéodory metric coincides with the majorizing Teichmüller-Kobayashi metric. This is an analogue for T of the corresponding Kra's result for the Abelian- Teichmüller disks in finite-dimensional Teichmüller spaces. III. \textit{R. Kühnau} [Comment. Math. Helv. 61, 290-307 (1986; Zbl 0605.30023)] had established that if f is holomorphic in \({\bar \Delta}{}^*\) and in (*) the equality holds, then necessary \(f=w^{\mu}| \Delta^*\) with \(\mu =k{\bar \phi}/| \phi |\) where \(\phi\) has the zeros of even order only. In the paper the required order of the smoothness of the boundary curves is essentially decreasing and there is shown that such is valid at least for \(f\in C^{2,\alpha}({\bar \Delta}^*)\), \(0<\alpha <1\…
Mathematics Subject Classification ID: 30C55
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 32F45
zbMATH DE Number: 4142391
zbMATH Keywords: Grunsky coefficients
zbMATH Keywords: Grunsky inequality
zbMATH Keywords: quasiconformal extension
zbMATH Keywords: Carathéodory metric
zbMATH Keywords: Beltrami differential
zbMATH Keywords: Teichmüller-Kobayashi metric
author: Samuel L. Krushkal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4142391