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Proceedings of the American Mathematical Society

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

 

 

Variability regions for bounded analytic functions with applications to families defined by subordination


Authors: Yusuf Abu-Muhanna and Thomas H. MacGregor
Journal: Proc. Amer. Math. Soc. 80 (1980), 227-233
MSC: Primary 30C75; Secondary 30D50
DOI: https://doi.org/10.1090/S0002-9939-1980-0577749-0
MathSciNet review: 577749
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Abstract: We examine the set of points $ (\varphi (\zeta ),\varphi '(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))$ where $ \vert\zeta \vert < 1$ and $ \varphi $ varies over the class of functions analytic in the open unit disk and is either (1) uniformly bounded or (2) subordinate to a given univalent function. In each case boundary points of the set correspond to unique functions associated with finite Blaschke products. This yields information about the form of solutions to extremal problems over the classes, including the problem

$\displaystyle \max \operatorname{Re} F(\varphi (\zeta ),\varphi '(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))$

where $ \vert\zeta \vert < 1$ and F is analytic.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0577749-0
Keywords: Analytic function, bounded function, variability region, extreme point, coefficient region, finite Blaschke product, extremal problems, subordination, univalent function
Article copyright: © Copyright 1980 American Mathematical Society