On extreme points of families of analytic functions with values in a convex set
type: Item
instance of: scholarly article
title: On extreme points of families of analytic functions with values in a convex set
zbMATH Open document ID: 0603.30028
DOI: 10.1007/BF01163355
author: Julian Gevirtz
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173769
review text: Let \(\Delta\) denote the unit disc \(\{z: | z| <1\}\) and D denote a convex domain in the complex plane which contains 0 and which is not the whole plane. Let F denote the family of all analytic functions f on \(\Delta\) for which \(f(\Delta) \subset D\) and \(f(0)=0\). For such functions \(f(\zeta)=\lim_{r\to 1}f(r\zeta)\) exists for almost all \(\zeta\in \partial \Delta\). \textit{Y. Abu-Muhanna} and \textit{Th. MacGregor} [Math. Z. 176, 511-519 (1981; Zbl 0461.30018)] considered the problem of determining the set E(F) of extreme points of F. It follows from a result of \textit{D. J. Hallenbeck} and \textit{Th. H. MacGregor} [Pac. J. Math. 50, 455-468 (1974; Zbl 0258.30015)] that if D is not a half plane, then \(\{f\in F:\) \(f(\zeta)\in \partial D\) for almost all \(\zeta\in \partial D\}\subset E(F)\). In the case D is a strip, \textit{J. G. Milcetich} [Proc. Am. Math. Soc. 45, 223-228 (1974; Zbl 0296.30015)] proved that the previous inclusion is actually an equality. \textit{Y. Abu-Muhanna} and \textit{Th. H. MacGregor} [Math. Z. 176, 511-519 (1981; Zbl 0461.30018)] proved the inclusion is an equality whenever D is a wedge. They conjectured this equality holds when D is any (bounded or unbounded) convex polygon. In this paper, the author proves that the conjecture is false. He first proves that if D is not a half plane, a strip or a wedge then there exists \(f\in E(F)\) for which \(f(\zeta)\not\in \partial D\) on a subset of \(\partial \Delta\) of positive measure. A second example is then constructed so that \(f\in E(F)\) and \(f(\zeta) \not\in \partial D\) for almost all \(\zeta\in \partial \Delta\). The measure theoretic considerations in this second construction are involved and clever. Both constructions depend in an essential way on properties of the Poisson integral formula. The problem remains to characterize the set E(F) when D is a convex polygon. At this point, it is not clear how to formulate a reasonable conjecture. This excellent paper is commende…
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 3973254
zbMATH Keywords: convex domain
zbMATH Keywords: extreme points
zbMATH Keywords: Poisson integral formula
reviewed by: David J. Hallenbeck
MaRDI profile type: Publication
cites work: Extreme points of families of analytic functions subordinate to convex mappings
cites work: Subordination and extreme-point theory
cites work: Q3704135
cites work: Linear Methods in Geometric Function Theory
cites work: Q4098830
Recommended article: Q4844857
Recommended article: Q3826805
Recommended article: Classes of analytic functions subordinate to convex functions and extreme points
Recommended article: Variability regions for families of analytic functions
Recommended article: On extreme points of subordination families with a convex majorant
Recommended article: Q4007690
Recommended article: On Extreme Points of Families Described by Subordination
Recommended article: Extreme points of some classes of analytic functions with positive real part and a prescribed set of coefficients
Recommended article: A Note on Extreme Points of Subordination Classes
Recommended article: Extreme points of integral families of analytic functions
description: scientific article; zbMATH DE number 3973254