On the Fekete-Szegö problem for close-to-convex functions. II (Q1097364)

[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.