Sur allures de la densité de Poincaré et ses dérivées au voisinage d'un point frontière
type: Item
instance of: scholarly article
title: Sur allures de la densité de Poincaré et ses dérivées au voisinage d'un point frontière
zbMATH Open document ID: 0794.30035
author: Shinji Yamashita
published in: Kodai Mathematical Journal
publication date: 1994-03-14T00:00:00Z
review text: Suppose \(\Omega\) is a hyperbolic region in the complex plane \(\mathbb{C}\) and \(a\in\mathbb{C}\) is an isolated boundary point of \(\Omega\). The author investigates the behavior of the density \(\lambda_ \Omega(z)\) of the hyperbolic, or Poincaré, metric and certain of its partial derivatives in a punctured neighborhood of \(a\). For instance, if \(\Phi_ n(z)=| z-a|^ n\log| z-a|^{-1}\), then \(\Phi_ 1(z)\lambda_ \Omega(z)\to 1/2\) when \(z\to a\). Also, the author shows that the quantities \(\Phi_ 2(z)|{\partial\lambda_ \Omega(z)\over\partial z}|\), \(\Phi_ 3(z)|{\partial^ 2\lambda_ \Omega(z)\over\partial z^ 2}|\) and \(\Phi_ 3(z)|{\partial^ 2\lambda_ \Omega(z)\over\partial\overline z\partial z}|\) are all bounded as \(z\to a\). These orders of growth at an isolated boundary point are exact for the density of the hyperbolic metric on the punctured disk \(\{z:0<| z|<1\}\). In addition, the author considers the behavior of \(\lambda_ \Omega(z)\) near an arbitrary boundary point. Finally, the author obtains the exact growth rate for the radius of univalence of the hyperbolic metric near an isolated boundary point.
reviewed by: D. Minda
Mathematics Subject Classification ID: 30F45
zbMATH DE Number: 515595
zbMATH Keywords: growth rate
zbMATH Keywords: hyperbolic metric
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039787
OpenAlex ID: W2091084954
cites work: The Poincaré Metric on the Twice Punctured Plane and the Theorems of Landau and Schottky
cites work: The Schwarzian derivative and schlicht functions
cites work: Q4405418
cites work: The derivative of a holomorphic function and estimates of the Poincaré density
cites work: Q3991878
cites work: Bounded analytic functions and metrics of constant curvature on Riemann surfaces
DOI: 10.2996/KMJ/1138039787
description: scientific article; zbMATH DE number 515595