Phe-schwarzian Derivative And Inner Radius Of Univalency

In this paper, the inner radius of univalency for hyperbolic domains bypre-Schwarzian derivative is studied, and some properties for the norm of pre-Schwarzian derivative and inner radius are established. As an application, the boundsof inner radius for angular domains are obtained. And we will discuss about the innerradius of harmonic mapping in the plane.The inner radius of univalency is related to many questions in the geometry the-ory of functions and descripe the important geometry invariant of hyperbolicRiemann Surface. Many scholars are interesting in estimating the inner radius ofunivalency for some special region, but it is very difficult to get the precise value ofinner radius of univalency for a plane domain. In this dissertation, we will estimatethe inner radius of univalency defined by the pre-Schwarzian derivative.The study about inner radius of univalency for analytical function has been veryactive, and some concrete results are obtained. However, the study for the problem ofthe univalency for harmonic mapping in the plane is not very satisfactory. The normof the Schwarzian derivative is very important tool for study univalency for analyticfunction. We will have some discussions on norm of univalent harmonic mapping inthe plane.The article includes three chapters.Chapter l, Preface. This chapter is devoted to the exposition of the basic theoryof the inner radius of univalency in the plane as well the development and theresearch situation of the theory of the inner radius of univalency, pre-Schwarzianderivative, harmonic mapping in the plane and its Schwarzian derivative. The mainresults of this dissertation are briefly introduced in this chapter.Chapter2, The inner radius of univalency for hyperbolic domains by pre-Schwarzian derivative. In this chapter, the inner radius of univalency for hyperbolicdomains by pre-Schwarzian derivative is studied, and some properties for the norm of pre-Schwarzian derivative and inner radius are established. As an application, thebounds of inner radius for angular domains are obtained.Chapter3, The norm of univalent harmonic mapping in the plane by Schwarzianderivative. Can we estimate the norm of univalent harmonic mapping on the plane asthe univalent conformal mapping? In this chapter, we mainly take some discussionsabout the Schwarzian norm of univalent harmonic mapping in the plane.