Univalent mappings associated with the Roper-Suffridge extension operator

The Roper-Suffridge extension operator provides a way of extending a (locally) univalent functionfεH(U) to a (locally) biholomorphic mappingF∈H(B_n). In this paper, we give a simplified proof of the Roper-Suffridge theorem: iff is convex, then so isF. We also show that iff∈S ^*, theF is starlike and that iff is a Bloch function inU, thenF is a Bloch mapping onB _n. Finally, we investigate some open problems. Partially supported by the Natural Sciences and Engineering Research Council of Canada under grant A9221.