The Generalized Roper-Suffridge Extension Operator On A Reinhardt Domain

In this thesis, the author mainly consider some properties of modified Roper-Suffridge extension operator in several complex variables. Some different condi-tions are given such that the extension operator preserves some known holomor-phic mappings, such as the almost starlike mappings of order α and the starlike mappings of order α. We also introduce a new subclass of starlike mappings SΩ*(β,A,β), which gcneralizes the spirallike mappings of type β and order α. and the strongly spirallike mappings of type β and order a from a unified perspectively.This thesis is divided into three chapters.In chapter1, we give a brief development of geometric function theory in several complex variables. Then we introduce some definitions, notations and the main results of this thesis.In chapter2, we generalize the Roper-Suffridge operator on Reinhardt do-mains. We prove that almost starlikeness of order α and starlikeness of order α are preserved under the generalized operator in some different conditions, respec-tively.In chapter3, we introduce a new subclass of starlike mappings SΩ*(β,A.B) on Reinhardt domains, which contains many known classes,such as the spirallike mappings of type β and order a, the strongly spirallike mappings of type β and order a. Further, we prove that the new subclass of starlike mappings is preserved under a generalized Roper-Suffridge extension operator on the Reinhardt domain The main results of this article generalize many known results. In particular, by making use of the Roper-Suffridge extension operator, we may easily give many concrete examples about these mappings. We unify some important subclasses of holomorphic mappings, which make geometric function theory of several complex variables much richer.