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

Abstract

Let $\widehat{\mathcal {S}}_g^{\alpha, \beta}(\mathbb{B}^n)$ be a subclass of normalized biholomorphic mappings defined on the unit ball in $\mathbb{C}^n,$ which is closely related to the starlike mappings. Firstly, we obtain the growth theorem for $\widehat{\mathcal {S}}_g^{\alpha, \beta}(\mathbb{B}^n)$. Secondly, we apply the growth theorem and a new type of the boundary Schwarz lemma to establish the distortion theorems of the Fr\'{e}chet-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.