Generalized Roper-Suffridge Operator for ϵ Starlike and Boundary Starlike Mappings

Abstract

This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties. First, we prove that the Roper-Suffridge extension operator preserves ϵ starlikeness on the open unit ball of a complex Banach space ℂ × X, where X is a complex Banach space. This result includes many known results. Secondly, by introducing a new class of almost boundary starlike mappings of order α on the unit ball Bn of ℂn, we prove that the Roper-Suffridge extension operator preserves almost boundary starlikeness of order α on Bn. Finally, we propose some problems.