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

London, R. R.

Proc. Am. Math. Soc. 117, No. 4, 947-950 (1993).

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.)

Keywords:

Fekete-Szegő problem; close-to-convex functions; starlike function

Citations:

Zbl 0635.30019; Zbl 0635.30020; Zbl 0635.35019