On the Fekete-Szegö problem for close-to-convex functions. II. (English) Zbl 0635.30020

Koepf, Wolfram

Arch. Math. 49, 420-433 (1987).

[For part I see Proc. Am. Math. Soc. 101, 89-95 (1987; reviewed above).]
Let C(\(\beta)\), \(\beta\geq 0\), denote the family of normalized close-to- convex function of order \(\beta\). (For \(\beta =1\) this is the usual class of close-to-convex functions as defined by Kaplan). In part I, the author found the sharp result for the functional \(| a_ 3-\lambda a^ 2_ 2|\) defined on the class C of Kaplan. In this paper, the author continues his investigations to the class C(\(\beta)\). We mention: Let \(f\in C(\beta)\), and let S(f) denote \[ S(f)=\sup_{z\in D}(1-| z|^ 2)^ 2| S_ f(z)|, \] (S\({}_ f(z)\) the Schwarzian derivative and D the unit disk) then \[ S(f)\leq\begin{cases} 2+4\beta,&\quad\text{if }\beta\leq 1\\ 2\beta^ 2+4&\quad\text{if }\beta\geq 1\end{cases} \] and the results are sharp.

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:

close-to-convex function; Schwarzian derivative

Citations:

Zbl 0635.30019

References:

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