Hyperbolically convex functions

Abstract

We investigate univalent holomorphic functions f defined on the unit disk D such that f(D) is a hyperbolically convex subset of D; there are a number of analogies with the classical theory of (euclidean) convex univalent functions. A subregion Ω of D is called hyperbolically convex (relative to hyperbolic geometry on D) if for all points a, b in Ω the arc of the hyperbolic geodesic in D connecting a and b (the arc of the circle joining a and b which is orthogonal to the unit circle) lies in Ω. We give several analytic characterizations of hyperbolically convex functions. These analytic results lead to a number of sharp consequences, including covering, growth and distortion theorems and the precise upper bound on |f ′′(0)| for normalized (f(0) = 0 and f ′(0) > 0) hyperbolically convex functions. In addition, we find the radius of hyperbolic convexity for normalized univalent functions mapping D into itself. Finally, we suggest an alternate definition of “hyperbolic linear invariance” for locally univalent functions f : D → D that parallels earlier definitions of euclidean and spherical linear invariance.