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.
