An improved Schwarz lemma at the boundary
type: Item
instance of: scholarly article
title: An improved Schwarz lemma at the boundary
zbMATH Open document ID: 1417.30016
author: Peter R. Mercer
published in: Open Mathematics
publication date: 2019-03-29T00:00:00Z
review text: Julia's classical lemma asserts that if $f$ is a holomorphic self-map of the unit disk with $f(1)=1$ (radial limit) and \[ \liminf_{z\to 1}\frac{1-|f(z)|}{1-|z|}=\beta\in\mathbb R, \] then $\beta>0$ and \[ \beta\geq\frac{|1-f(z)|^2}{1-|f(z)|^2}\;\frac{1-|z|^2}{|1-z|^2}. \] This classical result has recently been generalized or strengthened in various ways. \par The author uses another classical Schwarz-type lemma, Dieudonné's lemma, to prove the following improvement: \[ \beta\geq 2\,\frac{|1-f(z)|^2}{1-|f(z)|^2}\;\frac{1-|z|^2}{|1-z|^2}\;\frac{1-\text{Re}\left (f^*(z)\frac{1-\overline{f(z)}}{1-f(z)}\,\frac{1-z}{1-\bar{z}}\right )}{1-|f^*(z)|^2}, \] where $f^*(z)=\frac{1-|z|^2}{1-|f(z)|^2}\,f^\prime(z)$ (the hyperbolic derivative of $f$). The author also shows that the above estimate implies several of the known boundary Schwarz-type inequalities.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 7045495
zbMATH Keywords: Schwarz lemma
zbMATH Keywords: Julia lemma
MaRDI profile type: Publication
full work available at URL: math-2018-0096
OpenAlex ID: W2898867836
Wikidata QID: Q125028202
cites work: Q4888816
cites work: A sharp Schwarz inequality on the boundary
cites work: The Schwarz lemma at the boundary
cites work: Rigidity of Holomorphic Mappings and a New Schwarz Lemma at the Boundary
cites work: Q3660033
cites work: Some refined Schwarz-Pick lemmas
cites work: Sharpened versions of the Schwarz lemma
cites work: On angular derivative
cites work: Boundary distortion estimates for holomorphic maps
cites work: Boundary Schwarz lemma for holomorphic self-mappings of strongly pseudoconvex domains
cites work: The Schwarz inequality on the boundary for functions regular in the disk
DOI: 10.1515/MATH-2018-0096
description: scientific article; zbMATH DE number 7045495