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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Disproof of a coefficient conjecture for meromorphic univalent functions


Author: Anna Tsao
Journal: Trans. Amer. Math. Soc. 274 (1982), 783-796
MSC: Primary 30C50; Secondary 30C70
MathSciNet review: 675079
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Abstract: Let $ \Sigma$ denote the class of functions $ g(z) = z + {b_0} + {b_1}{z^{ - 1}} + \cdots $ analytic and univalent in $ \vert z\vert> 1 $ except for a simple pole at $ \infty$. A well-known conjecture asserts that $ \vert{b_n}\vert\, \leq 2/(n + 1)\qquad (n = 1,2, \ldots )$ with equality for $ g(z) = {(1 + {z^{n + 1}})^{2/(n + 1)}}/z = z + 2{z^{ - n}}/(n + 1) + \cdots $. Although the conjecture is true for $ n=1,2$ and certain subclasses of the class $ \Sigma$, the general conjecture is known to be false for all odd $ n\ge 3$ and $ n=4$.

In $ \S 2$, we generalize a variational method of Goluzin and develop second-variational techniques. This enables us in $ \S 3$ to construct explicit counterexamples to the conjecture for all $ n > 4$. In fact, the conjectured extremal function does not even provide a local maximum for $ {\text{Re}}\{ {b_n}\} $, $ n > 4$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0675079-6
PII: S 0002-9947(1982)0675079-6
Article copyright: © Copyright 1982 American Mathematical Society