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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On successive coefficients of univalent functions


Author: Ke Hu
Journal: Proc. Amer. Math. Soc. 95 (1985), 37-41
MSC: Primary 30C50
DOI: https://doi.org/10.1090/S0002-9939-1985-0796442-7
MathSciNet review: 796442
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Abstract: Let $ f(z) \in S$, that is, $ f(z)$ is analytic and univalent in the unit disk $ \left\vert z \right\vert < 1$, normalized by $ f(0) = f'(0) - 1 = 0$. Let $ p$ be real and

$\displaystyle {\{ f(z)/z\} ^p} = 1 + \sum\limits_{n = 1}^\infty {{D_n}(p){z^n}} .$

Lucas proved that

$\displaystyle \left\vert {{D_n}(p)} \right\vert - \left\vert {{D_{n + 1}}(p)} \... ...ert { \leq A{n^{(t(p) - 1)/2}}{{\log }^{3/2}}n,\quad n = 2,3, \ldots ,} \right.$

for some absolute constant $ A$ and $ t(p) = {(2\sqrt p - 1)^2}$. In this paper we improve $ t(p)$ as follows:

$\displaystyle T(p) = \frac{{4p - 1}}{{2p + t(p)}}t(p).$


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DOI: https://doi.org/10.1090/S0002-9939-1985-0796442-7
Article copyright: © Copyright 1985 American Mathematical Society