On the preservation of direction convexity under differentiation and integration (Q1768075)

A domain \(M\subset\mathbb{C}\) is convex in the direction \(e^{i\varphi}\) if for every \(a\in\mathbb{C}\) the set \(M\cap\{a+ te^{i\varphi}, t\in\mathbb{R}\}\) is either connected or empty. Let \(C(\varphi)\) denote the family of univalent analytic functions \(f\) in the unit disk \(D\) with \(f(0)= 0\) and \(f(D)\) convex in the direction \(e^{i\varphi}\). It was conjectured by Goodman and Saff and later proved by \textit{S. Ruscheweyh} and \textit{L. C. Salinas} [Ann. Acad. Sci. Fenn., Ser A I, Math. 14, No. 1, 63--73 (1989; Zbl 0674.30009)] that for \(0< r\leq\sqrt{2}- 1\) we have that \(f\in C(\varphi)\) but for \(\sqrt{2}- 1< r< 1\) this is not necessarily the case. In this paper the author investigates this property under differentiation and integration. The main result is that if \(f\in C(\varphi)\) then \(zf'(rz)\in C(\varphi)\) for \(0< r< 2-\sqrt{3}\) and the number \(2-\sqrt{3}\) is best possible.