Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Next Article Computational Methods and Function Theory 5 (2005), No. 1, 237--251 Copyright Heldermann Verlag 2005 A Modification of the Roper-Suffridge Extension Operator Jerry R. Muir, Jr. muirj2@scranton.edu , Department of Mathematics, University of Scranton, Scranton, PA 18510, U.S.A. [Abstract-pdf] [Abstract-ps] The Roper-Suffridge extension operator extends a locally univalent mapping defined on the unit disk of $\mathbb{C}$ to a locally biholomorphic mapping defined on the Euclidean unit ball of $\mathbb{C}^n$. Furthermore, the extension of a one variable mapping that is either convex or starlike has the analogous property in several variables. Motivated by recent results concerning the extreme points of the family $\mathcal{K}_n$ of normalized convex mappings of the Euclidean ball in $\mathbb{C}^n$, we introduce a new extension operator that, under precise conditions, takes the extreme points of $\mathcal{K}_1$ to extreme points of $\mathcal{K}_n$. In general, we examine the conditions under which this new extension operator will take a convex or starlike mapping of the unit disk to a mapping of the same type defined on the unit ball. Keywords: Roper-Suffridge extension operator, convex mapping, starlike mapping, extreme points. MSC 2000: Primary 32H02; Secondary 30C45. [FullText-pdf (266 K)] [FullText-ps (468 K)]