The extreme points of a class of functions with positive real part

Abstract

In spite of its elegance, extreme point theory plays a modest role in complex function theory. In a series of papers Brickman, Hallenbeck, Mac Gregor and Wilken determined the extreme points of some classical families of analytic functions. An excellent overview of their results is contained in [4]. Of fundamental importance is the availability of the extreme points of the set P of functions f analytic on the unit disc, with positive real part, normalized by f(0) = 1. These extreme points can be obtained from an integral representation formula given by Herglotz in 1911 [5]. A truly beautiful derivation of ExtP was given by Holland [6]. In this note we present yet another method, based on elementary functional analysis. As an application we determine the extreme points of the set F of functions f analytic on the unit disc, with imaginary part bounded by π 2 and normalized by f(0) = 0. They were originally determined by Milcetich [7] but our derivation is simpler. Finally we determine the extreme points of the set Pα of functions f ∈ P for which | arg f | ≤ απ2 for some constant α < 1. These were earlier described by Abu-Muhanna and Mac Gregor [1].