Roper-Suffridge Extension Operator And Loewner Chains

Loewner theory plays an important role in function theory of several complex vari-ables; and that Roper-Suffridge operator is vital in constructing the biholomorphic map-pings 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 relations between the operator and Loewner chains. 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 mapping of typeβand order a and al-most spirallike mappings of typeβand order a on domainsΩ′N={(z1,z2,…,zk)∈C×Cn2×…×Cnk:｜z1｜P1+‖z2‖2p2+…+‖zk‖kpk< 1, Pj≥1, j=1,2,…,k}.In the third chapter, we testify the generalized extension Roper-Suffridge operator can be embeded in Loewner chains on the unit ball in complex Banach spaces. At the same time, we obtain the fact that the operator keeps the properties of almost spirallike mapping of typeβand order a from the point of Loewner chains.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 Loewner chains.