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The extreme points of a class of functions with positive real part. (English) Zbl 0914.30013

Let \(H(\Delta)\) denote the set of analytic functions on the unit disk \(\Delta\), and let \(P\subset H(\Delta)\) consist of functions \(f\) with positive real part and normalized by \(f(0)= 1\). The extreme points of \(P\) are known. The author provides a different proof to determine the extreme points, which uses elementary functional analysis.
The set \(F\subset H(\Delta)\) consist of functions \(f\) with \(f(0)= 0\) and \(-\pi/2< \text{Im }f(z)< \pi/2\). The author determines the extreme points of \(F\) with a simpler proof than what is in the literature.
For \(0<\alpha< 1\), the set \(P_\alpha\subset P\) consists of functions that satisfy the inequality \(|\text{arg }f|< \alpha\pi/2\). The author finds the extreme points of \(P_\alpha\).

MSC:

30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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