On Refinements Of The Fekete-Szeg(?) Inequalities For The Subclasses Of Biholomorphic Mappings In Several Complex Variables

In this thesis,we mainly devote to studying Fekete-Szeg(?) ineqalities for the main subclasses of biholomorphic mappings in several complex variables,systematically.This paper is composed of four chapters.In chapter one,we briefly introduce the developmental background of FeketeSzeg(?) inequalities and some related problems.We also give some definitions and notations in this thesis.In chapter two,we mainly consider the order of zero,and obtain refinement of the Fekete-Szeg(?) inequality for a normalized convex function f on U.Then we extend this result to several complex variables,and establish refinement of the Fekete-Szeg(?)inequalities for quasi-convex mappings of type B(quasi-convex mappings of type A or quasi-convex mappings of type C).In chapter three,we firstly construct a widely representative class of holomorphic function on the unit disk,and then we establish the Fekete-Szeg(?) inequality by using the formula of Faà di Bruno for the higher order derivative of a composite function.Finally,we extend this result to higher dimensions.In chapter four,we mainly investigate the Fekete-Szeg(?) inequalities associated with spirallike mappings in one and higher dimensions,and these results contain some known results.The significance of the main work of this thesis lies in extending or improving some known results,and unifies many known results in form.