Distortion Theorems For Some Subclasses Of Biholomorphic Mappings Along A Unit Direction In Several Complex Variables

In this thesis, we study some subclasses of biholomorphic mappings whose distortion theorems are given in a new way and a whole new light in several complex, which further improve the geometric function theory. The thesis consists of five chapters.In the first chapter, we briefly introduce some definitions, notations, and the main results of this thesis.Starlike mapping in several complex variables is the generalization of starlike function in one complex variable. In chapter 2, we study the distortion theorems of some subclasses of starlike mappings along a unit direction on the unit polydisc in Cn which generalize the classical distortion theorem of starlike function on the disk that is the case n=1. Furthermore, the upper bound for them on the unit ball of a complex Banach space is obtained.Quasi-convex mappings only exist in several complex variables. In chapter 3, we give the distortion theorems of quasi-convex mappings along a unit direction on the unit polydisc in Cn which reduces to the classical distortion theorem of convex function on the disk as n=1. And, the upper bound for them on the unit ball of a complex Banach space is also achieved.In chapter 4, we discuss subclasses of starlike mapping deeply. The distortion theo-rems and the sharp distortion theorems for subclasses of starlike mappings along a unit direction on the unit polydisc in Cn are studied, which can reduce to the starlike mapping according to the value of a. In the meanwhile, we also obtain the upper bound for them on the unit ball of a complex Banach space.Spirallike mappings are more extensive than starlike mappings. In chapter 5, we obtain distortion theorems and the sharp distortion theorems for subclasses of spirallike mappings along a unit direction on the unit polydisc in Cn. And further, the upper bound for them on the unit ball of a complex Banach space is introduced.Owing to our research work, we have a further realization about the properties of the above subclasses of biholomorphic mappings in geometric function theory of several complex variable.