Almost Starlike Mappings Of Complex Order λ On The Unit Ball Of A Complex Banach Space

In this thesis, the author mainly considers some properties of almost starlike mappings of complex order A on the unit ball of a complex Banach space. Some different conditions are given such that the extension operator preserves some known holomorphic mappings, such as the almost starlike mappings of order a and the starlike mappings of order a. We also introduce an extended Roper-Suffridge in several complex variables on Reinhardt domains and prove that the extended Roper-Suffridge operator preserves almost starlike mappings under the certain conditions.This thesis is divided into three chapters.The first chapter, 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 establish some characterization of an almost starlike mappings of complex order λ from the point of the Loewner chains on the unit ball of a complex Banach space. Further, we draw a conclusion that the generalized Roper-Suffridge operator is embedded the Loewner chains on the different space and domains.In chapter3, we introduce a generalized Roper-Suffridge operator on Rein-hardt domains. At the same time, we prove that the new subclass of starlike mappings is preserved under a generalized Roper-Suffridge extension operator on the Reinhardt domainsThe 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.