Roper-Suffridge 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 mainly study the properties of the generalized Roper-Suffridge extensionoperator on special domains and the relations between the operator and Loewner chains.The whole thesis contains three chapters.In the first chapter, we briefly introduce the background of the development of thegeometric function theory in several complex variables, some notations, basic concepts,definitions and the main results of the thesis.In the second chapter, we respectively argue the generalized Roper-Suffridge exten-sion operator which preserves the property of almost spirallike mapping of typeβandorderαon the Reinhardt domainΩ_{p1,…,pn}={z∈Cⁿ:|z₁|^p₁+∑_jⁿ=2|z_j|^p_j<1,p₁∈(0,2],p_j≥1}and the unit ball in complex Hilbert space. Moreover, we briefly provethat the operator keeps spirallike mapping of typeβand orderαand almost spirallikemapping of typeβand orderαon the boundary complete Reinhardt domain.In the third chapter, first, we give a characterization of almost spirallike mapping oftypeβand orderα. in terms of Loewner chains. Next, we testify the generalized Roper-Suffridge extension operator can be embeded in Loewner chains on the unit ball in complexHilbert space, at the same time, we obtain the fact that the operator keeps the propertiesof 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 improvethem. So we have a deep realization about the Roper-Suffridge extension operator andLoewner chains.