Some Researches On Quasiconformal Mappings In The Plane

This paper consists of five chapters.In the first chapter, we briefly introduce the background of the problems which we will study; also we state the main results obtained in this paper.In the second chapter, by using Mori theorem, we study the quasiconformal mappings from the unit disk D = {z;|z|<1} to some special domains such as the upper half plane, the right half plane and the simply connected domain, and some new results about the distortion of f(x) are obtained.In the third chapter, based on the quasi-invariability and the monotonicity of the modulus of quasiconformal mappings, we discuss the extreme properties of quasiconformal mappings. We also give a new and simple proof to Koebe's 1/2-covering theorem for the case of the mappings being convex quasiconformal.In the fourth chapter, we study the Piranian and Weitsman's conjecture in the set of quasiconformal mappings. By using different methods we prove that the answer to this conjecture is positive.In the last chapter, we study the results obtained by Wu, Gehring and Martio etc about the quasicircles further, and some generalizations and a new necessary condition are obtained.