Convex And Starlike Mappings Associated With The Roper-Suffridge Extension Operator

In this thesis,the author mainly discusses some properties of generalized Roper-Suffridge extension operators in the field of multiple complex variables.Some different conditions are given such that the extension operator preserves some known holomor-phic mappings,such as the starlike mappings,convex mappings and the spirallike mappings of order ?,so on.The thesis is divided into three chapters.In Chapter 1,we briefly introduce the background of the development of geometric function theory of multiple complex variables,and some notations and definitions which used in the theorem,corollary and proof of this paper,as well as the narration of the main results.In Chapter 2,we generalize the Roper-Suffridge operator on Reinhardt domain,and extend the results on the unit disk to any given simply connected appropriate subdomain.Interestingly,we find that there seems to be no construction of Convex Mappings in Reinhardt domain.For this reason,we provide a new operator for the construction of Convex Mappings on unbounded domain in C2.Finally,by using the geometric method,we can get that if f is the spirallike function of order ?,then?N,1/p1,�1/pk(f)is the spirallike mapping of order ?.In Chapter 3,considering the failure of Riemann mapping theorem in higher dimensions,on the basis of Gong and Liu's mapping of convex functions on a unit disk using Caratheodory invariant metric in Reinhardt domain,the author mainly proves that Roper-Suffridge extension operator keeps convexity by proving some concavity of convex domain on C,and refines the construction of convex mapping.The main result of this paper is to generalize and improve the existing conclu-sions.Its significance lies in extending the special conclusion on the unit disk to the generalized simply connected subdomain,and constructing many new convex Map-pings on Reinhardt domain through Roper-Suffridge extension operator.Combining with geometric knowledge,the proof of some theorems is optimized,which makes the existing results richer and more beautiful.