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