Regional Pre-schwarz Derivative Of The Inner Radius Of Univalence,

The inner radius of univalency of plane domains by pre-Schwarzian derivative is studied in this paper,especially the lower bounds of inner radiuses for the domain bounded by a hyperbola and the outer domain of a triangle are obtained.There are three chapters in the thesis.The first chapter is the preface of this thesis.We introduce the theory of quasiconformal mappings,the theory of Teichm(u|¨)ller space and the latest developments of them.Furthermore,the problems discussed in this thesis and our main results are introduced.In chapter 2,we discuss the inner radius of univalency by pre-Schwarzian derivative,that is the problem to find the distance from a point in the Universal Teichm(u|¨)ller space embedded by pre-Schwarzian derivative to the boundary.We give an estimate of the lower bounds of inner radiuses for the domain bounded by a hyperbola and the outer domain of a triangle.In chapter 3,by use of the inner radiuses of univalency for strongly starlike domains,we give an estimate of the lower bounds of inner radiuses for some familiar domains.