Teichm¨¹ller Space Of The Inner Radius Of Univalence And Quasiconformal Dilatation And Regional,

This paper is concerned with the geometric property of the Universal Teichmüller space.Using the theory of Universal Teichmüller space,univalent functions,quasiconformal extension and the Loewner chain,we study the properties of different models of Universal Teichmüller space,explain the geometric meaning of the inner radius of univalency in the Pre-Schwarzian derivative model,and get some sufficient conditions for locally univalent and holomorphie(or meromorphic)functions to be wholly univalent in a given domain.There are five chapters in this thesis.The first chapter is the preface of it.We introduce the theory of quasiconformal mappings,the development of quasiconformal mapping and its application,the Universal Teichmüller space.Furthermore,the problems discussed in this thesis and our main results are introduced.In chapter two,we mainly study the geodesic property of the Universal Teichmüller Space and discuss whether the uniqueness of geodesics in different models of the Universal Teichmüller Space is equivalent.By construct a counterexample, we give a negative answer to this question,that is when there is a unique geodesic segment joining two points in one model of the Universal Teichmüller Space,while in other models,there may exist more than one geodesic segment joining the corresponding points.In chapter three,we discuss the inner radius of univalency by Pre-Schwarzian derivative of domains where the infinity is an inner point.By the definition and property of Universal Teichmüller Space,we get that the inner radius of univalency by Pre-Schwarzian derivative is the distance from a point in the Pre-Schwarzian derivative model of the Universal Teichmüller Space to its boundary.As an application, an estimation of the lower bound of inner radius of univalency by Pre-Schwarzian derivative of the infinitive domain bounded by an elliptic is obtained.In chapter four,by the theory of Loewner chain,we get some sufficient conditions for locally univalent and holomorphic(meromorphic) functions to be wholly univalent. We construct the quasiconformal extension with the Loewner chain and get the lower bound formula of the inner radius of univalency by Pre-Schwarzian derivative of the quasidisk domain. In chapter five,we discuss the problem of Pre-Schwarzian derivative and quasiconformal extension.This problem also relates to the Universal Teichmüller Space embedded by Pre-Schwarzian derivative.We find some connections between the complex dilatations of the quasiconformal extensions and the norms of the Pre-Schwarzian derivatives.Furthermore,we find another proof for the lower bound of the inner radius of univalency for angular domains by constructing an explicit quasiconformal extension of a class of holomorphic functions.