Regulated Domain And The Inner Redius Of Univlence

The present dissertation is concerned with the inner radius defined by the Schwarzianderivative or pre-Schwarzian derivative in the plane. The inner radius of univalence isrelated to many questions in the geometric theory of function and is used to descripethe important geometric invariant of hyperbolic Riemann surface. To estimate theinner radius of univalence of some special regions is an interesting problem for manyscholars, but it is very difficult to get the explicit value of inner radius of univalencefor a plane domain. In this dissertation, we estimate the inner radius of univalencedefined by the Schwarzian derivative or pre-Schwarzian derivative.Chapterâ… : Preface. This chapter is devoted to the exposition of the basic theoryof the inner radius of univalence in the plane, as well as the development and theresearch situation of the theory of the inner radius of univalence. The main results ofthis dissertation are briefly introduced in this chapter.Chapter â…¡: On the inner radius of univalence about classes of hyperbolic convexregion. In this chapter, we introduce some results about the inner radius of univalenceby Schwarzian derivative, then study the inner radius of univalence about classes ofhyperbolic convex region, finally it is turned to be a Nehari disk.Chapter â…¢: On the inner radius of univalence defined by the pre-Schwarzianderivative. In this chapter, making use of integral representation of a conformal mapfrom the unit disk to a regulated domain, we discuss the pre-Schwarzian derivative ofthe conformal map, then obtain some properties of the regulated domain, finally weobtain a new estimation of lower bound of the inner radius of univalence bypre-Schwarzian derivative of a regulated domain with convex corners.Chapter â…£: The Schwarzian derivative of the harmonic mappings in the plane.In this chapter, let h (z)be a conformal map of the unit disk onto regulated domain,by constructing a harmonic mapping f h gin the unit disk by harmonic shear, finally we get a boundary property of the Schwarzian derivative of f.