On successive coefficients of univalent functions
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- by Ke Hu PDF
- Proc. Amer. Math. Soc. 95 (1985), 37-41 Request permission
Abstract:
Let $f(z) \in S$, that is, $f(z)$ is analytic and univalent in the unit disk $\left | z \right | < 1$, normalized by $f(0) = fâ(0) - 1 = 0$. Let $p$ be real and \[ {\{ f(z)/z\} ^p} = 1 + \sum \limits _{n = 1}^\infty {{D_n}(p){z^n}} .\] Lucas proved that \[ \left | {{D_n}(p)} \right | - \left | {{D_{n + 1}}(p)} \right |\left | { \leq A{n^{(t(p) - 1)/2}}{{\log }^{3/2}}n,\quad n = 2,3, \ldots ,} \right .\] for some absolute constant $A$ and $t(p) = {(2\sqrt p - 1)^2}$. In this paper we improve $t(p)$ as follows: \[ T(p) = \frac {{4p - 1}}{{2p + t(p)}}t(p).\]References
- K. W. Lucas, On successive coefficients of areally mean $p$-valent functions, J. London Math. Soc. 44 (1969), 631â642. MR 243055, DOI 10.1112/jlms/s1-44.1.631 Hu Ke, On the coefficients of the starlike functions, J. Fudan Univ. 2 (1956), 77-81.
- W. K. Hayman, On successive coefficients of univalent functions, J. London Math. Soc. 38 (1963), 228â243. MR 148885, DOI 10.1112/jlms/s1-38.1.228
- M. Biernacki, Sur les coefficients tayloriens des fonctions univalentes, Bull. Acad. Polon. Sci. Cl. III. 4 (1956), 5â8 (French). MR 0076874 G. M. Goluzin, Method of variations in conformal mapping. II, Mat. Sb. (N.S.) 21 (1947), 83-115. (Russian)
Additional Information
- © Copyright 1985 American Mathematical Society
- 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