New subclasses of biholomorphic mappings and the modified Roper-Suffridge operator

Abstract

The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The Roper-Suffridge operator on the unit ball Bn in Cn is modified. By the analytical characteristics and the growth theorems of subclasses of spirallike mappings, it is proved that the modified Roper-Suffridge operator [ΦG,γ (f)](z) preserves the properties of SΩ*(A,B), as well as strong and almost spirallikeness of type β and order α on Bn. Thus, the mappings in SΩ*(A,B), as well as strong and almost spirallike mappings, can be constructed through the corresponding functions in one complex variable. The conclusions follow some special cases and contain the elementary results.