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A coefficient problem with typically real extremal function

Author: Seiji Konakazawa
Journal: Proc. Amer. Math. Soc. 123 (1995), 2723-2730
MSC: Primary 30C50; Secondary 30C70
MathSciNet review: 1257115
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Abstract: Let $ \Sigma_{0}$ denote the class of univalent functions in $ \vert z\vert > 1$, with expansion $ f(z) = z + \sum\nolimits_{n = 1}^\infty {{b_n}{z^{ - n}}} $. We show that if the omitted set of an $ f(z) \in {\sum _0}$ is on the trajectory arcs of the quadratic differential $ - w(w - \lambda )d{w^2}$ with $ \lambda \geqq 4(\sqrt 2 - 1)$, then $ f(z)$ has real coefficients. From this we can derive the coefficient estimate of $ {\max_{{\Sigma_{0}}}} \mathcal{R}e( - {b_3} - \frac{1}{2}b_1^2 + \lambda {b_2})$.

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

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Keywords: Univalent function, quadratic differential
Article copyright: © Copyright 1995 American Mathematical Society

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