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Proceedings of the American Mathematical Society

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

 

 

On extreme points of subordination families


Author: Yusuf Abu-Muhanna
Journal: Proc. Amer. Math. Soc. 87 (1983), 439-443
MSC: Primary 30C80
MathSciNet review: 684634
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Abstract: Let $ F$ be the set of analytic functions in $ U = \{ z:\vert z\vert < 1\} $ subordinate to a univalent function $ f$. Let $ D = f(U)$. For $ g(z) = f(\phi (z)) \in F$, let $ \lambda (\theta )$ denote the distance between $ g({e^{i\theta }})$ and $ \partial D$ (boundary of $ D$). We obtain the following results.

(1) If $ f'$ is Nevanlinna then $ \int_0^{2\pi } {\log \lambda (\theta )d\theta = - \infty } $ if and only if

$\displaystyle \int_0^{2\pi } {\log \left( {1 - \vert\phi ({e^{i\theta }})\vert} \right)d\theta = - \infty } .$

(2) If $ g$ is an extreme point of the closed convex hull of $ F$ then

$\displaystyle \int_0^{2\pi } {\log \left( {1 - \vert\phi ({e^{i\theta }})\vert} \right)d\theta = - \infty } ,$

for the case when $ D$ is a Jordan domain subset to a half-plane and $ f'$ is Nevanlinna.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0684634-5
Keywords: Analytic function, bounded function, convex function, extreme point, Jordan domain, Nevanlinna class, subordination, univalent function
Article copyright: © Copyright 1983 American Mathematical Society