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On the Fekete-Szegő theorem for close-to-convex functions. (English) Zbl 0922.30009

For \(0<\alpha\leq 1\) let \(S^*(\alpha)\) denote the class of normalised analytic functions \(f(z)\) in the open unit disc \(D\) of the complex plane satisfying \[ \left| \arg{zg'(z) \over g(z)}\right| <\alpha\pi/2\;(z\in D). \] Let \(K(\alpha)\) denote the class of normalised analytic functions \(f(z)\) in \(D\) for which there exists a \(g\in S^*(\alpha)\) such that \[ \text{Re} \bigl\{zf'(z)/g(z) \bigr\}>0. \] In this paper the authors obtain the sharp upper bounds for the functional \(| a_3-\mu a^2_2| (\mu\) real) where \(f(z)=z+a_2z^2+ a_3z^3+ \cdots\), belongs to \(K(\alpha)\). The upper bounds are too complicated to be reproduced here.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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