The Fekete-Szegö problem for strongly close-to-convex functions. (English) Zbl 0741.30008

The authors claim to extend work of the reviewer [Proc. Am. Math. Soc. 101, 89-95 (1987; Zbl 0635.30019)] about the Fekete-Szegö problem of maximizing the functional \(| a_ 3-\mu a^ 2_ 2|\quad(\mu\in\mathbb{R})\) for close-to-convex functions of order \(\beta\geq 0\). Unfortunately they do not work with the full class to close-to-convex functions \(f(z)=z+a^ 2_ 2+\ldots\) of order \(\beta\), for which there exists a starlike function \(g\) such that \[ \left|\arg(zf'(z)/g(z))\right|\leq{\pi\beta/2}, \] but they assume \(g\) to be normalized by \(g'(0)=1\). The resulting family is not linearly invariant [see e.g. the reviewer, Ann. Acad. Sci. Fenn., Ser. A I Math. 8, 349-355 (1983; Zbl 0508.30016)], i.e., it is not closed under the renormalized composition with an automorphism of the unit disk. For this restricted class considered the authors prove sharp estimates for the Fekete-Szegö problem when \(0\leq\beta\leq 1\), and for \(\mu\geq{2\over 3}\) when \(\beta>1\).
Reviewer: W.Koepf (Berlin)


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