Abstract: Let $D_{p_1,p_2,\cdots,p_n}=\{z\in \mathbb{C}^n: \sum\limits_{l=1}^n|z_l|^{p_l}<1\}, p_l> 1, l=1,2,\cdots,n$. In this article, we first establish the sharp estimates of the main coefficients for a subclass of quasi-convex mappings (including quasi-convex mappings of type $\mathbb{A}$ and quasi-convex mappings of type $\mathbb{B}$) on $D_{p_1,p_2,\cdots,p_n}$ under some weak additional assumptions. Meanwhile, we also establish the sharp distortion theorems for the above mappings. The results that we obtain reduce to the corresponding classical results in one dimension.

Key words: quasi-convex mapping, quasi-convex mapping of type $\mathbb{A}$, quasi-convex mapping of type $\mathbb{B}$, main coefficient, distortion theorem