Extension Operators Preserving Biholomorphic Mappings on Hartogs Domains

In this paper,the authors extend the Roper-Suffridge operator on the generalized Hartogs domains.They mainly research the properties of the extended operator.By the characteristics of Hartogs domains and the geometric properties of subclasses of spirallike mappings,they obtain the extended Roper-Suffridge operator preserving almost starlikeness of complex orderλ,almost spirallikeness of typeβand orderα,parabolic spirallikeness of typeβand orderρon the Hartogs domains in different conditions.They conclude that the corresponding extension operator preserves the same geometric invariance on the unit ball B^(n) in C^(n).The conclusions provide a new approach to study these geometric mappings in C^(n).