On Gehring-Pommerenke's estimate for quasicircles
type: Item
instance of: scholarly article
title: On Gehring-Pommerenke's estimate for quasicircles
zbMATH Open document ID: 0853.30014
author: Xinzhong Huang
published in: Kodai Mathematical Journal
publication date: 1996-12-11T00:00:00Z
review text: A Jordan curve \(\Gamma\) is called a quasicircle with constant \(M\) if \[ \min \{\text{diam} \Gamma_1, \text{diam} \Gamma_2\} \leq M |w_1 - w_2 |, \] where \(w_1, w_2 \in \Gamma\) and \(\Gamma_1, \Gamma_2\) are the two components of \(\Gamma \backslash \{w_1, w_2\}\). A domain bounded by a quasicircle with constant \(M\) is called a quasidisk with constant \(M\). Assume that \(f(z)\) is meromorphic and locally univalent in the unit disk \(D\), and \(|S_f |_D \leq b < 2\), where \(S_f\) is the Schwarzian derivative of \(f\). In this paper, the author proves that \(f(D)\) is a quasidisk with constant \[ M < 2 \left( 2 + \sqrt {3 + {b \over 6}} \right) \left( 1 - {b \over 2} \right)^{- {1 \over 2}}, \] which improves a result by \textit{F. W. Gehring} and \textit{Ch. Pommerenke} [Comment. Math. Helv. 59, 226-242 (1984; Zbl 0543.30014)], where the estimation for the constant \(M\) is \(M < 8(1 - {b \over 2})^{- {1 \over 2}}\).
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C75
zbMATH DE Number: 895927
zbMATH Keywords: quasicircle
zbMATH Keywords: Schwarzian derivative
zbMATH Keywords: quasidisk
reviewed by: Jixiu Chen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043543
OpenAlex ID: W2092009808
cites work: Quasiconformal reflections
cites work: A Uniqueness Theorem for Beltrami Equations
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cites work: Uniform domains and the ubiquitous quasidisk
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cites work: Q4334264
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DOI: 10.2996/KMJ/1138043543
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description: scientific article; zbMATH DE number 895927