Growth and Distortion Results for a Class of Biholomorphic Mapping and Extremal Problem with Parametric Representation in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^n$$\end{document}

Let S^gα,β(Bn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\mathcal {S}}_g^{\alpha , \beta }(\mathbb {B}^n)$$\end{document} be a subclass of normalized biholomorphic mappings defined on the unit ball in Cn,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^n,$$\end{document} which is closely related to the starlike mappings. Firstly, we obtain the growth theorem for S^gα,β(Bn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\mathcal {S}}_g^{\alpha , \beta }(\mathbb {B}^n)$$\end{document}. Secondly, we apply the growth theorem and a new type of the boundary Schwarz lemma to establish the distortion theorems of the Fréchet-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.