Abstract
The Roper-Suffridge extension operator extends a locally univalent mapping defined on the unit disk of ℂ to a locally biholomorphic mapping defined on the Euclidean unit ball of ℂn. 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 K n of normalized convex mappings of the Euclidean ball in ℂn, we introduce a new extension operator that, under precise conditions, takes the extreme points of K 1 to extreme points of 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.
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Muir, J.R. A Modification of the Roper-Suffridge Extension Operator. Comput. Methods Funct. Theory 5, 237–251 (2005). https://doi.org/10.1007/BF03321096
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DOI: https://doi.org/10.1007/BF03321096