One property of Schwarzian derivative
type: Item
instance of: scholarly article
zbMATH Open document ID: 0990.30013
published in: Demonstratio Mathematica
publication date: 2002-08-06T00:00:00Z
Mathematics Subject Classification ID: 30C55
zbMATH DE Number: 1619024
zbMATH Keywords: univalence
zbMATH Keywords: Schwarzian derivative
zbMATH Keywords: Poincaré metric
zbMATH Keywords: catastropic
MaRDI profile type: Publication
full work available at URL: dema-2001-0109
OpenAlex ID: W817552880
DOI: 10.1515/DEMA-2001-0109
title: One property of Schwarzian derivative
author: I. R. Kayumov
review text: Let \(D\) be a domain in the extended complex plane \(\overline{\mathbb{C}}\) with a quasiconformal boundary and \(A(D)\) be the set of analytic functions \(f: D \rightarrow \overline\mathbb{C}\). A functional \(I(f)\) is called admissible if there exists a positive number \(t_1\) such that \(f\in A(D)\) and \(I(f)\leq t_1\) implies \(f\) is univalent in \(D\). An admissible function \(I\) is called regular if there exists a positive \(t_n>\sup t_1\) such that \(f\in A(D)\) and \(I(f) \leq t_n\) implies \(f\) is no more than \(n\)-valent; otherwise \(I\) is called catastropic. The author has proved that the Schwarzian derivative \(S(f,z)=|(f''/f')'-(f''/f')^2/2|\) is catastropic in \(B_k=\{z\); \(|\arg z|< k\pi/2\}\) for \(0 < k < 2\).
reviewed by: T. N. Shanmugam
description: scientific article; zbMATH DE number 1619024