Coefficient inequalities for conformal mappings with homeomorphic continuation
type: Item
instance of: scholarly article
title: Coefficient inequalities for conformal mappings with homeomorphic continuation
zbMATH Open document ID: 0569.30016
DOI: 10.1007/BF00968961
published in: Siberian Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: The author proves several coefficient inequalities for functions in the class S, for Bieberbach-Eilenberg functions (i.e., functions analytic in the unit disk and with \(f(z_ 1)f(z_ 2)\neq 1\) for \(| z_ 1| <1\), \(| z_ 2| <1)\), and for Gelfer functions (i.e., functions of the form \(\phi (z)=1+\sum^{\infty}_{n=1}b_ nz^ n\) with \(\phi (z_ 1)+\phi (z_ 2)\neq 0\) for \(| z_ 1| <1\), \(| z_ 2| <1)\). Most of the theorems involve functions with homeomorphic extensions. An example of one of the shorter theorems has to suffice: Let \(f(z)=z+c_ 2z^ 2+...\in S,\) and suppose q(w) is analytic on the set \(d=C[f(| z| <1)\}.\) The coefficient \(a_ n\) \((n=\pm 1,\pm 2,...)\) are determined from the expansion \(q(f(z))=\sum^{\infty}_{n=- \infty}a_ nz^ n,\) which is valid in some annulus \(r<| z| <1\). Then the inequality \[ \sum^{\infty}_{n=1}n| a_ n|^ 2\leq K^ 2_ s(q/f)\sum^{\infty}_{n=1}n| a_{-n}|^ 2 \] holds. A necessary and sufficient condition for equality is also given.
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C75
zbMATH DE Number: 3908782
zbMATH Keywords: pairs of functions with nonoverlapping values
zbMATH Keywords: Bieberbach-Eilenberg functions
zbMATH Keywords: Gelfer functions
author: Arcadii Z. Grinshpan
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00968961
OpenAlex ID: W2053424297
description: scientific article; zbMATH DE number 3908782