Generalized Grunsky coefficients and inequalities
type: Item
instance of: scholarly article
title: Generalized Grunsky coefficients and inequalities
zbMATH Open document ID: 0629.30019
DOI: 10.1007/BF02766219
author: Reuven Harmelin
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let \(\alpha =\{\alpha_ n\}^{\infty}_{n=1}\) be a sequence of complex numbers and \[ \alpha (w)^{\ell}=(\sum^{\infty}_{n=1}\alpha_ nw^ n)^{\ell}=\sum^{\infty}_{k=1}A_{k,\ell}(\alpha)w^ k, \] \(A_{k,\ell}(\alpha)\) are the Bell polynomials. In this article the author investigates the connection between the Bell polynomials and some invariants introduced by the reviewer. These invariants are defined by the generating function \[ f'(z)/(f(z+w)- f(w))=(1/w)+\sum^{\infty}_{n=0}\phi_ n(w)z^ n, \] where \(\{\phi_ j\}^{\infty}_{j=1}\) are known to be invariant under Möbius transformation and \(\phi_ 1\) is the Schwarzian derivative multiplied by a constant. Further the author uses these investigations to improve some of the reviewer's results concerning the univalence of a meromorphic function as well as conditions for quasiconformal extension.
Mathematics Subject Classification ID: 30C55
Mathematics Subject Classification ID: 30C70
Mathematics Subject Classification ID: 30C50
zbMATH DE Number: 4023644
zbMATH Keywords: Aharonov invariants
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Bell polynomials
zbMATH Keywords: Möbius transformation
zbMATH Keywords: Schwarzian derivative
zbMATH Keywords: quasiconformal extension
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4023644