On The Inner Radius Of Univalence On The Plane

The present dissertation is concerned with the discussion of the inner radius of univalence on the plane and the related topics: Schwarzian derivative and the pre-Schwarzian derivative.The inner radius of univalence is related to many questions in the geometry theory of functions and descripe the important geometry invariant of hyperbolic Riemann Surface.Estimating the inner radius of univalence of some special regions is interesting problem for many scholars, but it is very difficult to get the precise value of inner radius of univalence for a plane domain. In this dissertation, we will estimate the inner radius of univalence defined by the Schwarzian derivative or the pre-Schwarzian derivative.ChapterⅠ:Preface. This chapter is devoted to the exposition of the basic theory of the inner radius of univalence on the plane, as well the development and the research situation of the theory of the inner radius of univalence. The main results of this dissertation are briefly introduced in this chapter.ChapterⅡ:On the inner radius of univalence about the trapezoid. We will take advantage of method of the David Calvis mainly to discuss the inner radius of univalence of the Isosceles trapezoid and the Right angle trapezoid, and get the conclusions as follows:Suppose P is an open Isosceles trapezoid with side sequence aaab and the smallest angle kπ(where b=a + 2acoskπ,0≤k≤1/3), then the inner radius of univalence isσ(P)=2k~2.Suppose P is an open Right angle trapezoid with side sequence aabc and the smallest angle (?)π(where b=2a,c=(?)a),then the inner radius of univalence isσ(P)=(?).ChapterⅢ:On the inner radius of univalence denned by the pre-Schwarzian derivative.What the condition of a locally injective analytic function is injective?The answer is related to the pre-Schwarzian derivative. In this chapter, we will discuss mainly the inner radius of univalence defined by the pre-Schwarzian derivative and get some conclusions.