Extremal Problems Of Quasiconformal Mappings And Schwarzian Derivatives

The present Ph.D. dissertation is concerned with the extremal problems in the theory of quasiconformal mappings and the related topics: quasiconformal extensions and Schwarzian derivatives.Quasiconformal mapping, which was posed by Grotzsch in 1928, is the generalization of conformal mapping in the theory of complex analysis. During the several decades, with the development of its theory, it has been widely spread into many research fields such as physics, science and technology, engineering, and other branches in mathematics, and provide a powerful tool for the study and research in these fields.The theory of extremal quasiconformal mappings is mainly concerned with the problems of existence and uniqueness of extremal quasiconformal mappings with given boundary correspondence and of the properties and characteristics of extremal quasiconformal mappings. Among which the problem of the characteristics of uniquely extremal quasiconformal mappings is the most difficult one and is most widely concerned. We discuss these problems in the second and third chapters of this paper, and obtain a series of deep results.The theory of Schwarzian derivatives has great significance in determining whether a conformal mapping has quasiconformal extensions, in estimating the inner radius of uni-valence of a domain and in discussing the properties of some conformal mapping families. The study of these key problems will be very important to the development of the theory of quasiconformal mappings. In the fourth and fifth chapters of this paper, we discuss the Schwarzian derivatives of analytic functions, the Nehari families and the extremal set of Schwarzian derivatives, and apply the obtained results to determine the inner radius of univalence of rectangles and hexagons with equal angles.Chapter I, Preface. This chapter is devoted to the exposition of the basic theory of quasiconformal mappings, of the development and the research situation of the theory of extremal quasiconformal mappings and the theory of Schwarzian derivatives (including Nehari families and the extremal set of Schwarzian derivatives). The main results of this Ph.D. dissertation are briefly introduced in this chapter.Chapter II, The characteristics of uniquely extremal quasiconformal mappings. In the family of quasiconformal mappings with given boundary correspondence, the extremal mapping must exist, but may be not unique. When is the extremal mapping unique, or what is the characteristic of uniquely extremal quasiconformal mapping is always the keyproblem. In this chapter, we first recall the development and the research situation of the theory of uniquely extremal quasiconformal mappings, mainly introduce and analyse the significant results obtained by Bozin V., Lakic N., Markovic V. and Mateljevic M.[14] in 1998. Then we study the characteristics of uniquely extremal quasiconformal mappings, and obtain some criterions of uniquely extremality which are different from the results of [14].Chapter III, Moduli of quadrilaterals and substantial points. In the theory of quasiconformal mappings, it is often difficult to evaluate the maximal dilatation of the extremal quasiconformal mapping. How to overcome the obstacle is also a hot point. According to the quasi-invariance of the moduli of quadrilaterals under quasiconformal mappings, it is natural to think of approximating the maximal dilatation of the extremal quasiconformal mapping by the ratios of the moduli of quadrilaterals. A key problem is: is it true that the supremum of the ratios of the moduli of quadrilaterals equals the maximal dilatation of the extremal quasiconformal mapping? In this chapter, firstly, we apply a result of [20] to prove that for a class of quasisymmetric homeomorphisms with substantial boundary points, the maximal dilatation of the extremal quasiconformal extension equals the supremum of the ratios of the moduli of quadrilaterals, which improve the result of [148]. Secondly, we prove that the above conclusion is also true for the boundary correspondence of afBne str...