Biholomorphic Mappings and the Extension Operators on New Hartogs Domains

In this paper, we generalize the Roper-Suffridge operator on the extended Hartogs domains. By using the geometric properties and the growth theorems of subclasses of biholomorphic mappings, we obtain the generalized operators preserve the properties of parabolic and spirallike mappings of type β and order ρ, S_Ω^* (β, A, B), almost starlike mapping of complex order λ on Ω_N under different conditions, and thus we get the corresponding results on the unit ball Bⁿ in Cⁿ. The conclusions lead to some known results.