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Proceedings of the American Mathematical Society
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
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:

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


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

  • [1] K. W. Lucas, On successive coefficients of areally mean 𝑝-valent functions, J. London Math. Soc. 44 (1969), 631–642. MR 0243055 (39 #4379)
  • [2] Hu Ke, On the coefficients of the starlike functions, J. Fudan Univ. 2 (1956), 77-81.
  • [3] W. K. Hayman, On successive coefficients of univalent functions, J. London Math. Soc. 38 (1963), 228–243. MR 0148885 (26 #6382)
  • [4] M. Biernacki, Sur les coefficients tayloriens des fonctions univalentes, Bull. Acad. Polon. Sci. Cl. III. 4 (1956), 5–8 (French). MR 0076874 (17,957f)
  • [5] G. M. Goluzin, Method of variations in conformal mapping. II, Mat. Sb. (N.S.) 21 (1947), 83-115. (Russian)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0796442-7
PII: S 0002-9939(1985)0796442-7
Article copyright: © Copyright 1985 American Mathematical Society