Two-point distortion theorems for univalent functions
The authors 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 \textit{Ch. Blatter} [Comment. Math. Helv. 53, 651-659 (1978; Zbl 0398.30004)]. 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. The authors obtain analogous results for convex univalent functions and convex regions, except that in this context the two-point comparison theorems do characterize convex regions. These results are all proved by an extension of Blatter's method which involves an interplay between differential geometry and univalent function theory.
scientific article; zbMATH DE number 537377