2-POINT DISTORTION-THEOREMS FOR UNIVALENT-FUNCTIONS

We establish a one-parameter family of symmetric, linearly invariant two-point distortion theorems for univalent functions defined on the unit disk. The weakest theorem in the family is a symmetric, linearly invariant form of a classical distortion theorem of Koebe, while another special case is a distortion theorem of Blatter. All of these distortion theorems are necessary and sufficient for univalence. Each of these distortion theorems can be expressed as a two-point comparison theorem between euclidean and hyperbolic geometry on a simply connected region; however, none of these comparison theorems characterize simply connected regions. We obtain analogous results for convex univalent functions and convex regions, except that in this context the two-point comparison theorems do characterize convex regions.