Punishing factors for angles
type: Item
instance of: scholarly article
title: Punishing factors for angles
zbMATH Open document ID: 1072.30014
DOI: 10.1007/BF03321030
author: Karl-Joachim Wirths
author: F. G. Avkhadiev
published in: Computational Methods and Function Theory
publication date: 2004-11-01T00:00:00Z
review text: The authors study the problem of maximizing \(\frac{f^{n}(z_0)}{n!}\) over all not necessarily univalent analytic functions \(f\) mapping some simply connected domain on the Riemann sphere into another, in terms of the Poincaré densities of \(z_0\) in the source domain and \(f(z_0)\) in the target domain, and sometimes also the Poincaré distance from \(z_0\) to \(\infty.\) Here's one of the main results: Suppose that \(f\) maps a simply connnected domain \(\Omega\) in the sphere into some sector \(H\) of given angular opening \(\alpha \pi,\) where \(1\leq \alpha \leq 2.\) Then extremal functions for the \(n\)th derivative problem are furnished by univalent maps of the sphere minus a plane line segment onto \(H.\) Tools in the paper include Taylor series of particular functions, subordination, the validity of Rogosinski's generalization of Bieberbach's conjecture \((| a_n| \leq n\,\) for every function in the unit disk which is subordinate to a function in \(S),\,\) and a generalization to meromorphic functions of Löwner's theorem about coefficients of inverse functions, due to G. Schober and the reviewer.
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C55
zbMATH DE Number: 2112931
reviewed by: Albert Baernstein II
MaRDI profile type: Publication
cites work: Q5179675
cites work: Q5673143
cites work: Schwarz-Pick inequalities for derivatives of arbitrary order
cites work: Schwarz-Pick inequalities for hyperbolic domains in the extended plane
cites work: Estimates for inverse coefficients of univalent functions from integral means
cites work: Q5673142
cites work: A proof of the Bieberbach conjecture
cites work: Q5574105
cites work: On the Coefficients of Subordinate Functions
cites work: Q3704141
cites work: Univalent functions. Selected topics
cites work: Higher derivatives of holomorphic function with positive real part
full work available at URL: https://doi.org/10.1007/bf03321030
OpenAlex ID: W2022637508
description: scientific article; zbMATH DE number 2112931