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Transactions of the American Mathematical Society

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



On bounded univalent functions whose ranges contain a fixed disk

Author: Roger W. Barnard
Journal: Trans. Amer. Math. Soc. 225 (1977), 123-144
MSC: Primary 30A32
MathSciNet review: 0422599
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Abstract: Let $ \mathcal{S}$ denote the standard normalized class of regular, univalent functions in $ K = {K_1} = \{ z:\vert z\vert < 1\} $. Let $ \mathcal{F}$ be a given compact subclass of $ \mathcal{S}$. We consider the following two problems. Problem 1. Find $ \max \vert{a_2}\vert$ for $ f \in \mathcal{F}$. Problem 2. For $ \vert z\vert = r < 1$, find the $ \max \;(\min )\vert f(z)\vert$ for $ f \in \mathcal{F}$. In this paper we are concerned with the subclass $ \mathcal{S}_d^\ast(M) = \{ f \in \mathcal{S}:{K_d} \subset f(K) \subset {K_M}\} $. Through the use of the Julia variational formula and the Loewner theory we determine the extremal functions for Problems 1 and 2 for the class $ \mathcal{S}_d^\ast(M)$, for all d, M such that $ \tfrac{1}{4} \leqslant d \leqslant 1 \leqslant M \leqslant \infty $.

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Keywords: Univalent, starlike, distortion theorems, variational techniques
Article copyright: © Copyright 1977 American Mathematical Society