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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On bounded univalent functions whose ranges contain a fixed disk
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by Roger W. Barnard PDF
Trans. Amer. Math. Soc. 225 (1977), 123-144 Request permission

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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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 225 (1977), 123-144
  • MSC: Primary 30A32
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0422599-3
  • MathSciNet review: 0422599