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Distortion and coefficient estimation of schlicht functions


Author: Ke Hu
Journal: Proc. Amer. Math. Soc. 87 (1983), 487-492
MSC: Primary 30C45; Secondary 30C50
DOI: https://doi.org/10.1090/S0002-9939-1983-0684644-8
MathSciNet review: 684644
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Abstract | References | Similar Articles | Additional Information

Abstract: Let

$\displaystyle S(k) = \left\{ {f(z) = z + \sum\limits_{n = 2}^\infty {{a_n}{z^n}... ...= \rho } \left\vert {f(z)} \right\vert = {\alpha _f} \geqslant k > 0} \right\}.$

In this paper, we prove that for $ f(z) \in S(k)$ the inequality

$\displaystyle \left\vert {{a_n}} \right\vert < n\sqrt {\frac{{P + \sqrt {(1 + p){A^{1/2}} - p} }}{{1 + P}}} $

holds where $ p = {k^2}/(1 - {k^2})$ and $ 1 \leqslant A < {(1.0657)^8}$. This strengthens a recent result of Horowitz.

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

  • [1] D. Horowitz, A further refinement for coefficient estimates for univalent functions, Proc. Amer. Math. Soc. 71 (1978), 217-220. MR 0480979 (58:1126)
  • [2] C. H. Fitzgerald, Quadratic inequalities and coefficient estimates for schlicht functions, Arch. Rational Mech. Anal. 46 (1972), 356-368. MR 0335777 (49:557)
  • [3] Hu Ke, Jiangxi Shiyuah Xue Bao 2 (1979), 9-16.
  • [4] G. M. Goluzin, Math Sb. 19 (61) 2 (1946), 203-236. MR 0018752 (8:325c)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0684644-8
Keywords: Univalent functions, schlicht functions, coefficients, quadratic inequalities, distortion
Article copyright: © Copyright 1983 American Mathematical Society

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