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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 $K = {K_1} = \{ z:|z| < 1\}$. Let $\mathcal {F}$ be a given compact subclass of $\mathcal {S}$. We consider the following two problems. Problem 1. Find $\max |{a_2}|$ for $f \in \mathcal {F}$. Problem 2. For $|z| = r < 1$, find the $\max \;(\min )|f(z)|$ 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