A Class Of Holomorphic Mappings And Research On The Properties Of Holomorph Functions

This thesis is concerned with some properties of some subclasses of holomorphic functions. This paper is composed of three chapters.In chapter one, we briefly introduce the developmental background of holomorphic function, and some definitions, notations in this thesis.In chapter two, we extend the Fekete-Szeg(?) problem for a class of normalized convex functions in the unit disk of one complex variable to several complex variables. In other words, we investigate the Fekete-Szeg(?) problem for a subclass of quasi-convex mappings of type B defined on the unit ball in a complex Banach space or on the unit polydisk in C~n.In chapter three, we derive sufficient condition for a certain general class of Carath(?)odory functions in the open unit disk by using Miller and Mocanu's lemma,and the results include and generalize many previous conclusions.The main result of this thesis is the breadth and the deep into further study under some known result, we improve and generalize some former researches. Though work of this paper,we have further knowledge for some properties of some important subclasses of holomorphic function.