Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
MathSciNet review: 577749
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30C75, 30D50

Retrieve articles in all journals with MSC: 30C75, 30D50

Additional Information

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

American Mathematical Society