In this thesis, we study some classes of holomorphic mappings in several complex variables, and obtain the estimations of some order item coefficients of homogeneous expansion for starlike mappings, close-to-starlike mappings and spirallike mappings relative to A.

Geometry quality for these mappings can be obtained from these inqualities, and some sufficiency conditions for starlike mappings and spirallike mappings are given.

In this paper, the authors obtain the parametric representation of starlike mappings on the unit ball Bn in Banach spaces; As a direct application, the growth theorem for starlike mappings is set up, which generalizes the corresponding results for starlike mappings on the unit ball in Cn .

As its application, we obtain some sufficient conditions for starlike mappings and close-to-starlike mappings on bounded balanced domains in Cn.

In this paper, The authors define the family of ε starlike mappings, in purpose to treat the family of convex mappings and the family of starlike mappings as one family, and to understand the transition from one family to another.

Prom these, we may construct a lot of concrete examples about biholomorphic starlike mappings, biholomorphic convex mappings, biholomorphicεstarlike mappings on some domains in Cn, or complex Hilbert spaces, or complex Banach spaces from e starlike functions on the unit disc U in C.