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Let $\widehat{\mathcal {S}}_g^{\alpha, \beta}(\mathbb{B}^n)$ be a subclass ofnormalized biholomorphic mappings defined on the unit ball in $\mathbb{C}^n,$which is closely related to the starlike mappings. Firstly, we obtain thegrowth theorem for $\widehat{\mathcal {S}}_g^{\alpha, \beta}(\mathbb{B}^n)$.Secondly, we apply the growth theorem and a new type of the boundary Schwarzlemma to establish the distortion theorems of the Fr\'{e}chet-derivative typeand the Jacobi-determinant type for this subclass, and the distortion theoremswith $g$-starlike mapping (resp. starlike mapping) are partly established also.At last, we study the Kirwan and Pell type results for the compact set ofmappings which have $g$-parametric representation associated with a modifiedRoper-Suffridge extension operator, which extend some earlier related results.

Growth and Distortion Results for a Class of Biholomorphic Mapping and Extremal Problem with Parametric Representation in $\mathbb{C}^n$