Roper-Suffridge Extension Operator And The Growth And Covering Theorems For Parabolic Starlike Mappings Of Order ρ

The growth and covering theorems of biholomorphic mappings play an important role in function theory of several complex variables; and that Roper-Suffridge operator is vital in constructing the biholomorphic mappings of several complex variables from the biholomorphic functions of complex variable. In this article, we study mainly the properties of the generalized extension Roper-Suffridge operator on special domains and the growth and covering theorem of biholomorphic mappings which has equipped special geometry properties. The whole thesis contains three chapters.In the first chapter, we introduce briefly the background of the development of the geometric function theory in several complex variables, some notations, basic concepts, definitions and the main results of the thesis.In the second chapter, we argue respectively the generalized extension Roper-Suffridge operator preserving the property of spirallike mappings on complex Hilbert spaceΩn(p1,P2, 2,…n+1} and preserving the properties of spirallike mapping of typeβand order a; almost spirallike mapping of typeβand order a on Reinhardt domainsΩn,P1,…,Pn={z∈In the third chapter, we proved that the growth and covering theorem of parabolic starlike mappings of order p by the principle of subordination on the unit ball Bn.The main results of this thesis are based on the known results, but extend and improve them. So we have a deep realization about the Roper-Suffridge operator and the growth and covering theorem of starlike mappings and their subclasses.