A note on Schwarz's lemma
type: Item
instance of: scholarly article
title: A note on Schwarz's lemma
zbMATH Open document ID: 1472.30011
author: Javad Mashreghi
published in: Complex Analysis and its Synergies
publication date: 2021-07-12T00:00:00Z
review text: The author uses the classical ideas related to the Alhlfors-Schwarz lemma and involving conformal metrics \(\rho\) and their Gaussian curvature \(k_\rho\) to prove the following result: Suppose that \(\rho_1,\rho_2\) are two metrics with strictly negative curvature on a planar domain \(\Omega\). Assume that \(\rho_2\) is strictly positive and continuous on \(\Omega\), and that \(\rho_1/\rho_2\) attains its maximum inside \(\Omega\). Then for every \(z\in\Omega\), \[ \frac{\rho_1(z)}{\rho_2(z)}\leq \left \|\frac{k_{\rho_1}(z)}{k_{\rho_2}(z)}\right \|_\infty^{1/2}. \] The author gives an application of this result for extremal metrics on star-shaped domains.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 7370976
zbMATH Keywords: Schwarz's lemma
zbMATH Keywords: SK-metric
zbMATH Keywords: Laplacian
zbMATH Keywords: curvature
MaRDI profile type: Publication
full work available at URL: s40627-021-00077-w
OpenAlex ID: W3148112204
Wikidata QID: Q125002673
cites work: An Extension of Schwarz's Lemma
cites work: Q3616135
cites work: On a Class of Conformal Metrics
cites work: Another elementary approach to the theorems of landau, montel, picard and schottky
cites work: The strong form of Ahlfors' lemma
DOI: 10.1007/S40627-021-00077-W
description: scientific article; zbMATH DE number 7370976