GENERALIZED ROPER-SUFFRIDGE OPERATOR FOR ε STARLIKE AND BOUNDARY STARLIKE MAPPINGS

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 epsilon starlikeness on the open unit ball of a complex Banach space C x 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 alpha on the unit ball B-n of C-n, we prove that the Roper-Suffridge extension operator preserves almost boundary starlikeness of order alpha on B-n. Finally, we propose some problems.