In the paper we study the class of univalent hyperbolically convex (h-convex) functions in the unit disc . Two types of normalization are considered. The class K h(α) is defined by the inner normalization . Another natural normalization is the so-called boundary normalization. We consider all h-convex functions f with fixed distance >0 from the boundary of to the origin. This class is denoted by H(c). The main result for both normalizations is the sharp upper estimate of . For the boundary normalization we use the extremal length method and derive some general two-point distortion theorems. For the class K h(α) we use the fact that the map f:2/z is starlike in .