ON THE DERIVATIVE OF HYPERBOLICALLY CONVEX FUNCTIONS

Abstract. A conformal mapping f of the unit disk D into itself is called hyperbolically convex if the non-euclidean segment between any two points of f(D) also belongs to f(D). In this paper we obtain the exact order of growth for the derivative of these functions and investigate their boundary behaviour. 1. The growth of the derivative Let D be the unit disk and T = @D. The analytic univalent function f: D! D is called hyperbolically convex (or simply h-convex) if the non-euclidean segment between any two points of f(D) also belongs to f(D). An h-convex function is continuous in D. Hyperbolically convex functions were ¯rst systematically studied by William Ma and David Minda [MM1]. Among many other results they obtained the char-acterization (1:1) Re 1 + z f 00(z) f 0(z