On hyperbolically convex functions

Abstract

Let\(\mathbb{D}\) be the unit disk of the complex plane. A conformai map of\(\mathbb{D}\) into itself is called hyperbolically convex if the non-Euclidean segment between any two points of\(f(\mathbb{D})\) also belongs to\(f(\mathbb{D})\). In this paper we prove several inequalities that are analogous to inequalities about (Euclidean) convex univalent functions. We show that if ƒ (0) = 0, then Re zf′/f > 1/2. This inequality is the key for the results of this paper. In particular we deduce a three-variable inequality corresponding to that of Ruscheweyh and Sheil-Small. The sharp bound for the Schwarzian derivative remains open.