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Extreme points of some classes of analytic functions with positive real part and a prescribed set of coefficients. (English) Zbl 0706.30013

Let \(P\) denote the set of functions \(f(z)=1+a_ 1z+a_ 2z^ 2+...\). that are analytic in the unit disc and satisfy Re\(f(z)>0\) for \(| z| <1\). Let \(B_ n=\{b_ 1,b_ 2,...,b_ n:| b_ k| <1\), \(k=1,2,...,n\}\) where \(n\) is a natural number, and let \(P(B_ n)=\{f\in P:\) \(a_ k=2b_ k\), \(k=1,...,n\}\). We prove that the set of extreme points of \(P(B_ n)\) consists exactly of the functions of the form \[ \sum^{2n+1}_{j=1}\lambda_ j\frac{1+x_ jz}{1-x_ jz}, \] where \(| x_ j| =1\), \(\lambda_ j\geq 0\). \(j=1,...,2n+1\), \(\sum^{2n+1}_{j=1}\lambda_ j=1\) and \(\sum^{2n+1}_{j=1}\lambda_ jx^ k_ j=b_ k\), \(k=1,...,n\).
Reviewer: D.P.Bellamy

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Keywords:

extreme points
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