Univalent Function?harmonic Measure And Asymptotically Conformal Extension

In this paper,we discuss the quasiconformal extension of univalent function in the unit disk(or outside the unit disk)and characterization of asymptotically conformal curve by harmonic measure.First,we study Loewner theory.Then,using Loewner theory,we give a sufficient condition for univalent function in the unit disk which can be asymptotically conformal extended to the whole plane,Second,we characterize the asymptotically conformal curve by harmonic measure.Last,we discuss some sufficient conditions for asymptotically conformal extension of univalent function with the form f(z)=z+?(z)or f(z)=z+?(1/z).There are four chapters in the thesis.In the first chapter,We will give a brief introduction to the quasiconformal mapping,and describe the problems studied and conclusions gotten in this paper.In chapter 2,we discuss univalent function by Loewner theory.Becker gives a sufficient condition for quasiconformal extensions of univalent function in the unit disk.Similarly,we give a sufficient condition for asymptotically conformal extensions of univalent function in the unit disk.Furthermore,we apply the obtained result to some special univalent functions(such as convex funcion,close-to-convex function)and give a sufficient condition for asymptotically conformal extensions of the subclass.In chapter 3,we use harmonic measure to characterize the asymptotically confor-mal curve.By using harmonic measure,Hag characterizes quasicircle,similar to the result of Hag,we use harmonic measure to characterize the asymptotically conformal curves.In chapter 4,we study the asymptotically conformal extension of univalentfunction with the form f(z)=z+?(z)(or f(z)=z + ?(1/z))and its generalization.