The branched Schwarz lemma: A classical result via circle packing
type: Item
instance of: scholarly article
title: The branched Schwarz lemma: A classical result via circle packing
zbMATH Open document ID: 0865.30034
author: Tomasz Dubejko
author: Kenneth Stephenson
published in: Michigan Mathematical Journal
publication date: 1997-07-17T00:00:00Z
review text: The seminal paper concerning the connection with circle packings of domains in the plane and analytic functions is that of \textit{B. Rodin} and \textit{D. Sullivan} [J. Differ. Geom. 26, 349-360 (1987; Zbl 0694.30006)] in which they proved a conjecture of W. Thurston on the approximation of conformal mappings via circle packings. Here, the authors develop aspects of a ``discrete analytic function theory'' based on circle packings to parallel classical function theory. In particular, they develop analogues in terms of circle packings of the following classical result (see, for example, \textit{Z. Nehari} [Duke Math. J. 14, 1035-1049 (1947; Zbl 0029.12302)]). (1) The branched Schwarz's lemma: Let \(f\), \(b\) be analytic functions mapping the unit disc \({\mathbf D}\) into itself, where \(f(0)=b(0)=0\), \(b\) is a finite Blaschke product, and the branch points of \(b\) form a subset (taking account of multplicity) of those of \(f\). Then \(|f'(0)|\leq|b'(0)|\); if \(|b'(0)|\neq 0\), equality holds iff \(f=\lambda b\) for some unimodular constant \(\lambda\). (2) The branched distortion lemma: Let \(f\) be an analytic function in \({\mathbf D}\) with \(f(0)=0\), \(r>0\), \(\Omega\) the component of \(f^1(r{\mathbf D})\) that contains \(0\), and \(f|_\Omega:\Omega\to r{\mathbf D}\) a proper mapping. Let \(b\) be a finite Blaschke product whose branch points consist of the branch points of \(f\) that lie in \(\Omega\) (taking account of multiplicity), and with \(b(0)=0\). Then \(|f'(0)|\geq r|b'(0)|\); if \(|b'(0)|\neq 0\), equality holds iff \(f=(\lambda r)b\) for some unimodular constant \(\lambda\). The authors assert that the discrete theory inspires and helps to prove classical results, and look forward to new ``classical'' results turning up in this way. First the authors review the definitions and notation associated with circle packings. They then formulate and prove discrete versions of the standard Schwarz and distortion lemmas, along with a number of related …
Mathematics Subject Classification ID: 30C80
Mathematics Subject Classification ID: 30C20
zbMATH DE Number: 838727
zbMATH Keywords: circle packings
zbMATH Keywords: Schwarz's lemma
reviewed by: David A. Brannan
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005225
OpenAlex ID: W2031214576
Wikidata QID: Q124986816
DOI: 10.1307/MMJ/1029005225
description: scientific article; zbMATH DE number 838727