The Study Of Biholomorphic Mappings And The Corresponding Operators In Several Complex Analysis

The theory of several complex variables derives from one complex variable,yet they are essentially different.Biholomorphic mappings are the main research objects in several complex variables.In order to generalize the theory of one complex variable to higher-dimensional complex spaces,we need discuss biholomorphic mappings which have special geometric properties,such as starlike mappings and complex mappings.We have lots of research results about starlike mappings and convex mappings,while we have known few subclasses and expansions of these two classes of mappings.Moreover,the properties of various subclasses of biholomorphic mappings will change with the spaces and domains discussed.So it is quite necessary to discuss the properties of biholomorphic mappings which have special geometric properties.It is an important research subject in geometric function theories of several complex variables to construct biholomorphic mappings which have special geometric properties in higher-dimensional complex spaces.The Roper-Suffridge operator has put up a bridge between the theories of one complex variable and several complex variables.Applying the operator we can construct the corresponding biholomorphic mappings of several complex variables by some biholomorphic functions of one complex variable with special geometric properties.But the existing Roper-Suffridge extension operators may only preserve parts of subclasses and expansions of biholomorphic mappings.Subclasses of biholomorphic mappings with various geometric properties are constantly emerging,so we need study extensions of the Roper-Suffridge operator and the properties of the extended operators to preserve various biholomorphic mappings.The theory of holomorphic functions is not only applied in other fields of mathe-matics,but also it is an important tool to study Dynamics,Physics and other subjects.In the process of application,scholars found that we need discuss the broader function classes in some cases,such as polyanalytic functions.The research results of polyanalyt-ic functions have been very rich in the theory of one complex variable,while relatively few in several complex variables.So in this paper we study k-holomorphic functions in higher-dimensional complex spaces.Cauchy integral formulas and the corresponding Cauchy singular integrals play an important role in boundary-value problems of analytic functions.There have been rel-atively perfect results about Riemann-Hilbert boundary-value problems in one complex variable,while relatively few in several complex variables.In this paper we discuss the properties of Cauchy singular integral operators with k-holomorphic kernels and study some boundary-value problems of k-holomorphic functions in spaces of several complex variables.This paper consists of four chapters.In the introduction,the background,curren-t research status of several complex variables and the main results of this paper are introduced.In chapter 1,from the geometric properties of conic domains,we define new subclass-es of starlike functions(spirallike functions)-k-conic starlike functions of order α(k-conic spirallike functions of type β and order α),generalize the definition to spaces of sev-eral complex variables and define new subclasses of starlike mappings-k-conic starlike mappings of order α.Applying the principle of subordination,we discuss the coefficient estimation.Fekete-Szego inequality of k-conic starlike functions of order α,k-conic spi-rallike functions of type β and order α on the unit disc and k-conic starlike mappings of order α on bounded starlike and circular domains.We obtain the growth,covering and deviation theorems of k-conic starlike mappings of order α on Bn in Cn.In chapter 2,we extend the Roper-Suffridge operator to the generalized Hartogs domains.Applying the geometric properties of various subclasses of biholomorphic map-pings,we detailedly study the geometrical invariability of the extended operators that they preserve SΩ*(β A,B),strong and almost spirallikeness of type β and order α,parabol-ic spiralikeness of type β and order ρ on Hartogs domains on the different conditions.Sequentially,we get the properties of the corresponding extension operators on Bn.In chapter 3,from k-holomorphic functions of one complex variable,we define k-holomorphic functions in Cn,study their properties and obtain some conclusions parallel to the properties of holomorphic functions.We mainly discuss Cauchy integral theorem,Cauchy integral formula of k-holomorphic functions in Cn and a series of inferences:mean value theorem,Cauchy inequality,uniqueness theorem,Taylor theorem,Laurent theorem,Liouville theorem,Weierstrass theorem and so on.In chapter 4,from the Cauchy integral formula of k-holomorphic functions on bi-cylinder domains,we define the Cauchy singular integral and the Cauchy principal value with k-holomorphic kernel.We study the properties of Cauchy singular integral opera-tors with k-holomorphic kernel and obtain the corresponding Plemelj formula.Applying the Plemelj formula and the boundary properties of Cauchy singular integrals,we study the boundary-value problems of k-holomorphic functions on bicylinder and generalized bicylinder domains.We discuss the existence and the integral expression of solutions to boundary-value problems.