Generalized Schwarz lemmas for meromorphic functions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1255.30032
DOI: 10.4134/BKMS.2012.49.2.417
author: Xiaojun Huang
author: Ling Chen
published in: Bulletin of the Korean Mathematical Society
publication date: 2012-04-13T00:00:00Z
full work available at URL: http://www.mathnet.or.kr/mathnet/kms_content.php
Mathematics Subject Classification ID: 30C80
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 6023971
zbMATH Keywords: generalized Schwarz lemma
zbMATH Keywords: argument principle
zbMATH Keywords: winding number
zbMATH Keywords: homotopy
MaRDI profile type: Publication
Wikidata QID: Q124818853
title: Generalized Schwarz lemmas for meromorphic functions
review text: The authors give two generalized Schwarz lemmas which apply to meromorphic functions in the unit disc \(\mathbb{D}\). In Theorem 1 it is assumed that \(f\) is meromorphic in \(\mathbb{D}\) with a zero of order \(k\) at 0, that there is a number \(r_0\in (0,1)\) such that \(|f(z)|\leq1\) for \(r_0<|z|<1\), and that, if \(f^{(k)}(0)\neq0\), the function \(g(z):=z^k-k!f(z)/f^{(k)}(0)\) satisfies \(Z(g)-P(g)\neq k\), where \(Z(g)\) and \(N(g)\) denote, respectively, the number of zeroes and poles of \(g\) in \(\mathbb{D}\). Under these assumptions it is then proved that NEWLINE\[NEWLINE\frac{1}{k!}|f^{(k)}(0)|\leq1,NEWLINE\]NEWLINE with equality if and only if \(f(z)=cz^k\) for some constant \(c\) of modulus 1.NEWLINENEWLINE In Theorem 2 it is assumed again that \(f\) is meromorphic in \(\mathbb{D}\) with a zero of order \(k\) at 0, that there is a number \(r_0\in (0,1)\) such that \(|f(z)|\leq1\) for \(r_0<|z|<1\), and that, for \(\omega \in \mathbb{D}\) not a pole of or a zero of \(f\), the function \(g_\omega(z):=z^k-\omega^kf(z)/f(\omega)\) satisfies \(Z(g_\omega)-P(g_\omega)\neq k\). In this case, the conclusion is that NEWLINE\[NEWLINE|f(\omega)|\leq|\omega|^k,NEWLINE\]NEWLINE with equality if and only if \(f(z)=cz^k\) for some constant \(c\) of modulus 1.NEWLINENEWLINEExamples are given to show that the last condition in each of the theorems is necessary. The proofs are applications of the argument principle.
reviewed by: Line Baribeau
description: scientific article; zbMATH DE number 6023971