On the mean value theorem for analytic functions
type: Item
instance of: scholarly article
title: On the mean value theorem for analytic functions
zbMATH Open document ID: 0608.30024
author: Julian Gevirtz
published in: Michigan Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: The author looks for a certain analogue of the mean value theorem for analytic functions defined in the unit disk \({\mathbb{D}}\) of the complex plane \({\mathbb{C}}\). Therefore, he defines for some region \(R\subset {\mathbb{C}}\) the family B(R) of analytic functions on \({\mathbb{D}}\) with f'(\({\mathbb{D}})\subset R\), and m(R) to be the supremum of all \(r<1\) for which \(f\in B(R)\) implies that \[ (f(b)-f(a))/(b-a)\in R,\quad | a|,| b| <r. \] One gets easily that \(m(R)=1\) if and only if R is convex, i.e. in this case the situation is quite similar to the real one. The author gives a necessary and sufficient condition on R for \(m(R)>0\). He shows that for R simply connected m(R)\(\geq 2-\sqrt{3}\) (which is the radius of convexity for univalent functions). In the further paper a variational method is developed which gives some information about the nature of a function \(f\in B(R)\) for which m(R) is actually the ''mean value radius''.
reviewed by: W. Koepf
Mathematics Subject Classification ID: 30C70
zbMATH DE Number: 3983564
zbMATH Keywords: Bloch constant
zbMATH Keywords: variation of extremal functions
zbMATH Keywords: universal covering surface
zbMATH Keywords: mean value theorem
zbMATH Keywords: radius of convexity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003416
OpenAlex ID: W2061247273
DOI: 10.1307/MMJ/1029003416
description: scientific article; zbMATH DE number 3983564