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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On extreme points of families described by subordination


Author: Rahman Younis
Journal: Proc. Amer. Math. Soc. 102 (1988), 349-354
MSC: Primary 30C80,; Secondary 30D55
DOI: https://doi.org/10.1090/S0002-9939-1988-0920998-7
MathSciNet review: 920998
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Abstract: Let $ s\left( F \right)$ denote the set of analytic functions in $ D = \{ z:\vert z\vert < 1\} $ subordinate to an analytic function $ F$. It is shown that if $ F$ is a polynomial then the extreme points of the closed convex hull of $ s(F) \subset \{ F \circ \phi :\phi \in {\text{extreme}}\;{\text{points}}\;{\text{of}}\;B(H_0^\infty )\} $. Also if $ F(z) = {((z - \alpha )/(1 - \bar \alpha z))^n},\vert\alpha \vert < 1$ and $ n$ is a positive integer then the extreme points of the closed convex hull of $ s(F) = \{ F \circ \phi :\phi \in {\text{extreme}}\;{\text{points}}\;{\text{of}}\;B(H_0^\infty )\} $. An analogue of Ryff's theorem, and other results related to subordination in Bergman spaces have been obtained.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0920998-7
Keywords: Analytic functions, polynomials, extreme points, subordination, Bergman spaces
Article copyright: © Copyright 1988 American Mathematical Society