Sharpness of certain Campbell and Pommerenke estimates
type: Item
instance of: scholarly article
title: Sharpness of certain Campbell and Pommerenke estimates
zbMATH Open document ID: 0914.30014
DOI: 10.1007/BF02312838
published in: Mathematical Notes
publication date: 1999-06-23T00:00:00Z
review text: Let \(\Delta\) denote the open unit disk. A set \({\mathfrak M}\) of functions \(f(z)= z+\sum^\infty_{n= 2}a_n(f)z^n\) analytic in \(\Delta\) is called a linearly invariant family if for each \(f\in{\mathfrak M}\) the following two conditions are satisfied: (1) \(f'(z)\neq 0\) in \(\Delta\) and (2) for any conformal automorphism \(\varphi(z)\) of \(\Delta\), the function \({f(\varphi(z))- f(\varphi(0))\over f'(\varphi(0))\varphi'(0)}= z+\cdots\) also belongs to \({\mathfrak M}\). This notion was introduced by C. Pommerenke in 1964. Pommerenke defined the order of a linearly invariant family \({\mathfrak M}\) as \(\sup_{f\in{\mathfrak M}}| a_2(f)|\). The universal linearly invariant family \(U_\alpha\) is the union of all linearly invariant families \({\mathfrak M}\) such that the order of \({\mathfrak M}\) is at most \(\alpha\) (for \(\alpha\geq 1\)). The authors examine several estimates for \(|\text{arg }f'(z)|\) and determine extremal functions, where \(f\) belongs to some linearly invariant family. Other results in the paper include a lower estimate for the Schwarzian derivative for functions in \(U_\alpha\) and sharp upper bounds for the order of the functions \(f_r(z)= {f(rz)\over r}\), where \(f\in U_\alpha\).
reviewed by: Renate McLaughlin
Mathematics Subject Classification ID: 30C75
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 1262496
zbMATH Keywords: linearly invariant family
zbMATH Keywords: Schwarzian derivative
author: Janusz Godula
author: Viktor V. Starkov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02312838
OpenAlex ID: W2020064190
description: scientific article; zbMATH DE number 1262496