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Proceedings of the American Mathematical Society

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

 

 

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: 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.

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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