Hyperbolically Convex Functions
by
Christian Pommerenke
Technische Universitat Berlin
Coauthors: Diego Mejia (Universidad Nacional de Colombia Medellin)

A conformal map f of the unit disk D of the complex plane into itself is called hyperbolically convex if if the hyperbolic segment between any two points of f(D) also lies in f(D). These functions form a non-linear space invariant under Moebius transformations of D onto itself. The fact that this space is non-linear makes it impossible to use many of the standard methods.

This survey talk will concentrate on

- Analytic characterizations of h-convex functions

- Inequalities for h-convex functions

- Hausdorff dimension of image sets

A few proofs will be sketched.

Date received: August 3, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-21.