Conformal Mappings And The Hyperbolic Metrics

In this thesis, we study conformal mappings and the hyperbolic metric of rectangles. Besides,the hyperbolic convexity of conformal metric and Nehari class is also discussed.Conformal mapping is an important branch of functions of complex variable, it is the geometrical theory of analytic functions. With an intensive study of conformal mappings, it plays an important role in the fields of physics, technology and other branches of mathematics, it provides an useful research tool for the development of other subjects.The full thesis divides into three chapters.Chapter 1: Preface. This chapter is devoted to the exposition of the basic theory of conformal mappings and hyperbolic metric, including an introduction of the background of the problems which are discussed in this paper, besides, the main results of this thesis are also briefly introduced in this chapter.Chapter 2: On the hyperbolic metric of rectangles. By using the properties of elliptic function we get the hyperbolic metrics of a rectangle at four points around its center and the hyperbolic metrics of a kind of rectangle whose modulus is m;Besides, a formulae for the hyperbolic density of two lines of a rectangle is given, while, the hyperbolic geodesics of a rectangle are also discussed in the last section.Chapter 3: On the hyperbolic convexity of conformal metric and Nehari class. Denote the family of analytic functions satisfying Nehari's univalence criterion by Nehari class. In this chapter, we first introduce some basic theory of Schwarzian derivative and Nehari class, then we study the hyperbolic convexity of conformal metric and Nehari class whose Schwarzian derivatives satisfy |S_f(z)|≤2a(1+(1-a)|z|²)(1-|z|²)^-2,(1≤a≤2), and get a condition for the hyperbolic convexity of powers of the Poincare metric of the image.