Properties And Relations Of Subclasses Of Biholomorphic Mappings In Several Complex Variables

In this thesis, we study the properties and relations of subclasses of biholomorphic mappings in several complex variables systematically. It consists of four chapters.In the first chapter, we introduce some definitions, notations, and the main results of this thesis briefly.In Chapter 2, we introduce two classes of holomorphic mappings (M_g|^) and M_g on the unit ball in a complex Banach space and bounded starlike circular domain in C~n, respectively, and consider the order of zero (i.e., the origin 0 is a zero of order k + 1 of a mapping f(x) － x). The growth, covering theorems and the estimations of homogeneous expansion for the mapping f(x) are obtained. As corollaries, we unify all the related results about the class of starlike mappings and its subclasses, the class of spirallike mappings of type β and its subclasses. Especially, from the proofs of corollaries, we can not only see the essential relations of these mappings more clearly but also rationalization of the subclasses of spirallike mappings of type β which are introduced in [Feng1].In Chapter 3, we give a characterization of a subclass of biholomorphic mappings in terms of Loewner chains. Meanwhile, we also investigate the generalized Roper-Suffridge extension operators on two important classes of Reinhardt domains in C~n, and prove the property that they can be embedded in Loewner chains. As corollaries, we also obtain the results that they preserve the properties of starlikeness, spirallikeness of type β and almost starlikeness of order α. So, we obtain effective methods to construct the starlike mappings, the spirallike mappings of type β and almost starlike mappings of order α on two important classes of Reinhardt domains in C~n.In the last chapter, we study a subclass of quasi-convex mappings——Quasi-convex mappings of order a on the unit ball of a complex Banach space. The relations between Quasi-convex mappings of order α and convex mappings are given. the growth and covering theorems are established. Moreover, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order a defined on the polydisc in C~n and on the unit ball in a complex Banach space, respectively. As corollaries, we also obtain the growth and covering theorems, the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping.