Distortion and coefficient estimation of schlicht functions
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- Proc. Amer. Math. Soc. 87 (1983), 487-492 Request permission
Abstract:
Let \[ S(k) = \left \{ {f(z) = z + \sum \limits _{n = 2}^\infty {{a_n}{z^n} \in S,} \;{{\overline {\lim } }_{\rho \to {1^ - }}}\frac {{{{(1 - \rho )}^2}}}{\rho }\max \limits _{\left | z \right | = \rho } \left | {f(z)} \right | = {\alpha _f} \geqslant k > 0} \right \}.\] In this paper, we prove that for $f(z) \in S(k)$ the inequality \[ \left | {{a_n}} \right | < n\sqrt {\frac {{P + \sqrt {(1 + p){A^{1/2}} - p} }}{{1 + P}}} \] holds where $p = {k^2}/(1 - {k^2})$ and $1 \leqslant A < {(1.0657)^8}$. This strengthens a recent result of Horowitz.References
- David Horowitz, A further refinement for coefficient estimates of univalent functions, Proc. Amer. Math. Soc. 71 (1978), no. 2, 217–221. MR 480979, DOI 10.1090/S0002-9939-1978-0480979-0
- Carl H. Fitzgerald, Quadratic inequalities and coefficient estimates for schlicht functions, Arch. Rational Mech. Anal. 46 (1972), 356–368. MR 335777, DOI 10.1007/BF00281102 Hu Ke, Jiangxi Shiyuah Xue Bao 2 (1979), 9-16.
- G. Golusin, Method of variations in the theory of conform representation, Rec. Math. [Mat. Sbornik] N.S. 19(61) (1946), 203–236 (Russian, with English summary). MR 0018752
Additional Information
- © Copyright 1983 American Mathematical Society
- 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