Growth and Distortion Results for a Class of Biholomorphic Mapping and Extremal Problem with Parametric Representation in Cn

Let S alpha,beta(Bn) be a subclass of normalized biholomorphic mappings defined on the unit ball in Cn, which is closely related to the starlike mappings. Firstly, we obtain the growth theorem for Sg alpha,beta. Secondly, we apply the growth theorem and a new type of the boundary Schwarz lemma to establish the distortion theorems of the Frechet-derivative type and the Jacobi-determinant type for this subclass, and the distortion theorems with g-starlike mapping (resp. starlike mapping) are partly established also. At last, we study the Kirwan and Pell type results for the compact set of mappings which have g-parametric representation associated with a modified Roper-Suffridge extension operator, which extend some earlier related results.