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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Support points of families of analytic functions described by subordination

Authors: D. J. Hallenbeck and T. H. MacGregor
Journal: Trans. Amer. Math. Soc. 278 (1983), 523-546
MSC: Primary 30C45; Secondary 30C80
MathSciNet review: 701509
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Abstract: We determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination. The families we study include the functions with a positive real part, the typically-real functions, and the functions which are subordinate to a given majorant. If the majorant $ F$ is univalent then each support point has the form $ F \circ \;\phi $, where $ \phi $ is a finite Blaschke product and $ \phi (0) = 0$. This completely characterizes the set of support points when $ F$ is convex. The set of support points is found for some specific majorants, including $ F(z) = {((1 + z)/(1 - z))^p}$ where $ p > 1$. Let $ K$ and $ {\text{St}}$ denote the set of normalized convex and starlike mappings, respectively. We find the support points of the families $ {K^{\ast} }$ and $ {\text{St}}^{\ast} $ defined by the property of being subordinate to some member of $ K$ or $ {\text{St}}$, respectively.

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Keywords: Analytic function, continuous linear functional, support point, extreme point, closed convex hull, subordination, univalent function, convex mapping, starlike mapping, function with a positive real part, bounded function, Herglotz formula, probability measure, finite Blaschke product
Article copyright: © Copyright 1983 American Mathematical Society