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 order lambda, almost spirallikeness of type beta and order alpha, parabolic spirallikeness of type beta and order rho on 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 DOUBLE-STRUCK CAPITAL C-n. The conclusions provide a new approach to study these geometric mappings in DOUBLE-STRUCK CAPITAL C-n.