Univalent analytic functions and the Poincaré metric
type: Item
instance of: scholarly article
title: Univalent analytic functions and the Poincaré metric
zbMATH Open document ID: 0726.30037
author: Shinji Yamashita
published in: Kodai Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: The author investigates two types of problems on a hyperbolic region \(\Omega\) in the complex plane \({\mathbb{C}}\). The density \(\rho_{\Omega}\) of the Poincaré, or hyperbolic, metric plays a role in both types of problems. First, the author relates the geometry of \(\Omega\) to various differential inequalities involving the quantity \(1/\rho_{\Omega}\). Second, the author studies the family S(\(\Omega\)) of holomorphic, univalent functions defined on \(\Omega\). In particular, he extends a result of \textit{P. L. Duren}, \textit{H. S. Shapiro} and \textit{A. L. Shields} [Duke Math. J. 33, 247-254 (1966; Zbl 0174.375)] to hyperbolic regions of finite type. This result gives an upper bound for a norm of the Schwarzian derivative of \(f\in S(\Omega)\) in terms of a second norm related to \(f''/f'\). Also, the author shows that if a function f in S(\(\Omega\)) is ``isolated'' in a sense related to the second norm, then \({\mathbb{C}}\setminus \Omega\) has zero area. Reviewer's comments. Note that the regions of finite type in this paper are sometimes called uniformly perfect by other authors. Also, \textit{R. Harmelin} [Isr. J. Math. 70, 111-128 (1990; Zbl 0708.30045)] independently obtained certain results relating the geometry of a hyperbolic region to differential inequalities involving \(\rho_{\Omega}\). There is some material common to the two papers; anyone interested in the subject matter of one of these papers should certainly read the other.
reviewed by: D. Minda
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 30C55
zbMATH DE Number: 4198423
zbMATH Keywords: hyperbolic metric
zbMATH Keywords: Schwarzian derivative
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039216
OpenAlex ID: W1967118999
cites work: Q5179675
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DOI: 10.2996/KMJ/1138039216
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description: scientific article; zbMATH DE number 4198423