Estimates of spherical derivative of meromorphic functions
type: Item
instance of: scholarly article
title: Estimates of spherical derivative of meromorphic functions
zbMATH Open document ID: 1005.30022
author: Shinji Yamashita
published in: Hokkaido Mathematical Journal
publication date: 2003-02-05T00:00:00Z
review text: Let \(f\) be a meromorphic function on the unit disk \(D\). The author considers various analytic-geometric quantities related to \(f\). These quantities include: (i) The spherical derivative \(f^\sharp (z)= |f^\prime(z)|/(1+|f(z)|^2)\). (ii) The maximum \(\rho_a(z,f)\) of all \(r\in (0,1]\) such that \(f(w)\neq -1/\overline{f(z)}\), for all \(w\) in the hyperbolic disk \(\Delta(z,r)\) of center \(z\) and hyperbolic radius \(r\). (iii) \(\rho_a(f):= \inf\{\rho_a(z,f):z\in D\}\). (iv) The maximum \(\rho(z,f)\) of all \(r\in (0,1]\) such that \(f\) is univalent in \(\Delta(z,r)\). (v) The maximum \(\rho_\omega(z,f)\) of all \(r\in (0,1]\) such that \(e^{f(w)}+e^{f(z)}\neq 0\) for all \(w\in \Delta(z,r)\). (vi) \(\rho_\omega (f):= \inf\{\rho_\omega (z,f):z\in D\}\). (vii) The Schwarzian derivative of \(f\), and several other quantities. The author proves various sharp estimates involving the above quantities. Of particular interest is the investigation of the equality cases and the identification of the extremal functions. Among the main results of the paper are the following: (a) \(f\) is normal (that is, the function \((1-|z|^2) f^\sharp(z)\) is bounded in \(D\)) if and only if \(\rho_a(f)>0\). (b) A holomorphic function \(f\) on \(D\) is Bloch (that is, \(\sup_{z\in D}(1-|z|^2) |f^\prime(z)|<\infty\)) if and only if \(\rho_\omega(f)>0\).
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30F45
zbMATH DE Number: 1751377
zbMATH Keywords: meromorphic function
zbMATH Keywords: normal function
zbMATH Keywords: hyperbolic distance
zbMATH Keywords: Schwarzian derivative
zbMATH Keywords: Bloch function
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1350911775
OpenAlex ID: W2050195179
DOI: 10.14492/HOKMJ/1350911775
description: scientific article; zbMATH DE number 1751377