The derivative of a holomorphic function and estimates of the Poincaré density
type: Item
instance of: scholarly article
title: The derivative of a holomorphic function and estimates of the Poincaré density
zbMATH Open document ID: 0752.30005
author: Shinji Yamashita
published in: Kodai Mathematical Journal
publication date: 1993-01-16T00:00:00Z
review text: The author proves several inequalities for the derivative of a function holomorphic in the unit disc \(D\), generalizing classical inequalities, and gives applications to the Poincaré density of the image domain \(G=f(D)\) and the Schwarzian derivative of \(f\). From these results we quote Theorem 3. Let \(f\) be holomorphic in \(D\). Then \(\delta(z,f)\leq{4\rho(z,f)F(z,f)\over \{\rho(z,f)+1\}^ 2}\). Equality for some \(z=z_ 0\) with \(f'(z_ 0)\neq 0\) is possible only for one of the functions \(f(z)=p{z-z_ 0\over 1-\overline{z_ 0}z}+q\), \(p\neq 0\) and \(q\) constants. Here, \(F(z,f)=(1-| z|^ 2)| f'(z)|\), \(\delta(z,f)\) is the radius of the largest schlicht disc contained in the image domain (or Riemann surface) \(f(D)\) centred at \(f(z)\), and \(\rho(z,f)\) is the radius of the largest non-euclidean disc \(\left|{\zeta-z\over 1-\overline z\zeta}\right|<r\) centred at \(z\) where \(f\) is univalent.
Mathematics Subject Classification ID: 30C55
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 92935
zbMATH Keywords: derivative
zbMATH Keywords: Poincaré density
zbMATH Keywords: Schwarzian derivative
zbMATH Keywords: schlicht disc
zbMATH Keywords: non-euclidean disc
zbMATH Keywords: univalent function
reviewed by: Norbert Steinmetz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039530
OpenAlex ID: W2073147421
cites work: The Poincaré Metric of Plane Domains
cites work: Q4478672
cites work: Q4220704
cites work: The Poincaré Metric on the Twice Punctured Plane and the Theorems of Landau and Schottky
cites work: Remarks on a paper by Zeev Nehari
cites work: Q5666814
cites work: Two Theorems on “Schlicht” Functions
cites work: A distortion theorem for conformal mappings with an application to subharmonic functions
cites work: The Schwarzian derivative and schlicht functions
cites work: On univalent functions in the unit disk whose image contains a given disk
cites work: Linear-invariante Familien analytischer Funktionen. I
cites work: Problems and theorems in analysis II. Theory of functions, zeros, polynomials, determinants, number theory, geometry. Transl. from the German by C. E. Billigheimer.
cites work: On the Derivatives of Functions Analytic In the Unit Circle and Their Radii of Univalence and of p-Valence
cites work: Univalent analytic functions and the Poincaré metric
DOI: 10.2996/KMJ/1138039530
description: scientific article; zbMATH DE number 92935