Abstract: In this paper, we define the class $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$ of $g$-parametric starlike mappings of real order $\gamma$ 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 Fr\'{e}chet-derivative type of $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$ with a weak restrictive condition, we further obtain the distortion results of the Jacobi-determinant type and the Fr\'{e}chet-derivative type for the corresponding classes (compared with $\widehat{\mathcal {S}}^{\gamma}_g(\mathbb{B}_{\mathbb{X}})$) defined on the unit polydisc (resp. unit ball with the arbitrary norm) in the space of $n$-dimensional complex variables, $n\geqslant2$. 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.

Key words: Banach space, distortion theorem of Jacobi-determinanttype, distortion theorems of the Fréchet-derivative type, $g$-parametric starlike mappings