## Fekete-Szegö inequalities for close-to-convex functions.(English)Zbl 0771.30007

The author claims to extend work of the reviewer [Proc. Am. Math. Soc. 101, 89-95 (1987; Zbl 0635.35019) and Arch. Math. 49, 89-95 (1987; Zbl 0635.30020)] about the Fekete-Szegő problem of maximizing the functional $$| a_ 3-\mu a_ 2^ 2|$$ ($$\mu\in\mathbb{R}$$) for close- to-convex functions of order $$\beta\geq 0$$. Unfortunately he does not work with the full class of close-to-convex functions $$f(z)=z+a_ 2 z^ 2+\dots$$ of order $$\beta$$, for which there exists a starlike function $$g$$ such that $\left|\arg {{zf'(z)} \over {g(z)}}\right| \leq {{\pi\beta} \over 2}$ but he assumes $$g$$ to be normalized by $$g'(0)=1$$. For this restricted class he proves sharp estimates for the Fekete-Szegő problem.
Reviewer: W.Koepf (Berlin)

### MSC:

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

### Citations:

Zbl 0635.30019; Zbl 0635.30020; Zbl 0635.35019
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