Some Problems Of The Quasiconformal Mappings

The present Ph.D. dissertation is concerned with the extremal problems in the theory of quasiconformal mappings and the related topics: Schwarzian derivative, quasiconformal extension and John domain.Chapter I: Preface. This chapter is devoted to the exposition of the basic theory of quasiconformal mappings, of the development and the reseach situation of the theory of quasiconformal mappings and the theory of Schwarzian derivatives, including the analytic and geometric properties of functions in the Nehari class. The main results of this Ph.D. dissertation are briefly introduced in this chapter.Chapter II: On the distortion theorems and quasiconformal extensions of the Nehari class. Denote the family of analytic functions satisfying Nehari's univalence criteria by Nehari class. The researches on the theory of univalency criteria of analytic functions by Nehari and the theory of the quasiconformal extensions by Ahlfors and Weill revealed the deep connection between the Schwarzian derivative and the quasiconformal extension of univalent functions. In this chapter, we first analyse the relationship between Schwarzian derivative and the second order linear ordinary differential equation, and then by using the comparison theorems of ordinary differential equation, we study the distortion properties and quasiconformal extensions of a class of Nehari functions. In the last section, we construct an explicit quasiconformal extension of this class of Nehari function. Our work extended some results obtained by Chuaqui and Osgood, Gehring and Pommerenke, Ahlfors and Weill.Chapter III: John disks and the Schwarzian derivative. John disks can be thought as " one-sided quasidisks ", a Jordan domain Cl C C is a quasidisk if and only if and * = C \ are John disks. This chapter is concerned with functions in a subclass of the Nehari class whose Schwarzian derivatives satisfy (1 - |z|2)|Sf(z)| < 4. We discuss the connection between those functions and John disks. We also prove some results concerning a new function which is related to the Schwarzian derivative, logarithmic derivative and John disk. Then we give a new sufficient condition on for = f(D) to be a John disk.Chapter IV: Logarithmic derivative and quasiconformal extension. Using the relationship between the logarithmic derivative and the univalency of a function, we discuss the distortion properties and quasiconformal extension of a class of univalent functions, and then an explicit quasiconformal extension of this class is obtained.Chapter V: The exponent of convergence of quasiconformal Fuchsian groups. Using the quasi-invariant property of the hyperbolic distance under a quasiconformal mapping, we discuss the exponent of convergence of quasiconformal Fuchsian groups, and then we give an estimate related to a conjecture which is posed by Bonfert-Taylor and Taylor.Chapter VI: On the extremal set of extremal quasiconformal mapping. Let f(z) be an extremal quasiconformal mapping of the unit disk D onto itsef, X[p] - [z z G D, \n(z)\ - Halloo}, A = /*/~1- ^ there exists a compact set E C X[p] with mesE > 0, satisfyingthen there exists an extremal Beltrami coefficients v(z] ~ M(Z), which satisfies X[v\ C X[] -E, where E = f-1(E).We also obtain a similar result on infinitesimally extremal Beltrami coefficients. At the end, we obtain a sufficient condition for (Z) to be uniquely extremal.