Distortion Theorems for Classes of g-Parametric Starlike Mappings of Real Order in ℂⁿ

Abstract

In this paper, we define the class $\hat {\cal S}_g^\gamma ({\mathbb{B}_\mathbb{X}})$ of g-parametric starlike mappings of real order γ on the unit ball ${\mathbb{B}_\mathbb{X}}$ in a complex Banach space $\mathbb{X}$, where g is analytic and satisfies certain conditions. By establishing the distortion theorem of the Fréchet-derivative type of $\hat {\cal S}_g^\gamma ({\mathbb{B}_\mathbb{X}})$ with a weak restrictive condition, we further obtain the distortion results of the Jacobi-determinant type and the Fréchet-derivative type for the corresponding classes (compared with $\hat {\cal S}_g^\gamma ({\mathbb{B}_\mathbb{X}})$) defined on the unit polydisc (resp. unit ball with the arbitrary norm) in the space of n-dimensional complex variables, n ⩾ 2. Our results extend the classic distortion theorem of holomorphic functions from the case in one-dimensional complex space to the case in the higher dimensional complex space. The main theorems also generalize and improve some recent works.