Null-Homotopic Holomorphic Mappings In Several Complex Variables

Starlike mappings and spiral-like mappings are two of the most important mappingsin geometric function theory of several complex variables.Their same geometric character-ization is that the closed line segment or spiral joining each point in their image domainsto zero lies entirely in their image domains.In this thesis,we study the mappings havingthe above geometric characterization from the view of homotopy.The whole thesis consists of three chapters. In the first chapter,we introduce brie?ysome notations,basic concepts,definitions and theorems used usually in this thesis. Inchapter 2 ,we introduce the definition of null-homotopic holomorphic mappings,and studythe properties of null- homotopic holomorphic mappings on the bounded convex circulardomains in Cn.Moreover,we obtain the criteria for null- homotopic holomorphic mappingson the bounded convex circular domains in Cn.In chapter 3,we introduce the definition ofnull-homotopic holomorphic mappings in Banach spaces,and study the properties of null-homotopic holomorphic on the unit ball in complex Banach spaces.Moreover,we obtainthe criteria for null- homotopic holomorphic mappings on the unit ball in complex Banachspaces.The principal theorems are relative to the known theorems of starlike mappings andspiral-like mappings that have the above geometric characterization in geometric functiontheory of several complex variables.