Sharp distortion theorems for a subclass of quasi-convex mappings in several complex variables

In this paper, the sharp growth, covering theorems and the sharp distortion theorems of the Frechet derivative type for a subclass of quasi-convex mappings (include quasi-convex mappings of type and quasi-convex mappings of type ) on the unit ball of complex Banach spaces are first established. Meanwhile, the sharp growth, covering theorems and the sharp distortion theorems of the Frechet derivative type for the above generalized mappings on the unit polydisc in are given. Moreover, the refined distortion theorems of the Jacobi determinant type for a subclass of quasi-convex mappings (include quasi-convex mappings of type ) on the unit ball with arbitrary norm in are also obtained, and the refined distortion theorems of the Jacobi determinant type for the above generalized mappings on the unit polydisc in are derived. Our results generalize some known conclusions and reduce to the classical results in one complex variable.