## On the Fekete-Szegő problem for close-to-convex functions.(English)Zbl 0635.30019

Fekete-Szegö were first to prove that $$\max_{f\in S} | a_ 3- \lambda a^ 2_ 2| =1+2 \exp[-2\lambda(1-\lambda)],\lambda\in [0,1].$$ In this paper the author solves the similar question for the class of close-to-convex functions. In particular: $$| | a_ 3| -| a_ 2| | \leq 1$$ for the class of close-to-convex functions, while the result for the class S is $$\max_{s}| | a_ 3| -| a_ 2| | =1.029...$$ Among other means, the author uses the following result of the present author [Coefficients of symmetric functions of bounded boundary rotation (to appear)]. Lemma 1. Let f be a close to convex normalized function in the unit disc. Then h defined by $h'(z)=[f'(z^ 2)]^{1/2},\quad h(0)=0,$ is an odd close-to-convex function of order 1/2.
Reviewer: D.Aharonov

### MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C50 Coefficient problems for univalent and multivalent functions of one complex variable 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods

### Keywords:

Fekete-Szegö problem; close-to-convex functions

Zbl 0635.30020
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