The Inner Radius Of The Plane Domains

The inner radius is the important geometrical features of the universal Teichmuller space. It reflects the position of the analytic functions and its equivalence classes in the universal Teichmuller space, and it is one of the study object in which the complex analysises are very intrested. It is related to many questions of the geometric function theory. The studying of the inner radius is very active all the time. Many scholars such as Nehari,Hille,Lehtinen,Ahlfors,Gehring,Lehto,Calvis,Wieren and so on, come to the inner radius conclusions of a series of special areas. For example, the unit disk,the simply connected domain,regular n-side polygon,the side ratio of the rectangles is [1,1.52346…],equiangular hexagonal etc.This article researches the inner radius of parallelogram,isosceles trapezoid and equiangular octagon. This article is divided into three chapters: Chapter One, Preface. We introduce the basic theory of the quasiconformal mappings simply first of all. And then we review the development of the quasiconformal mappings and the Schwarzian derivative theory. Thirdly we descript the situation of the inner radius's study. At last we introduce our main task in this article . Chapter Two, The inner radius of plane domains. We divided it into three parts in this chapter. First, Parallelogram domain; Second, Isosceles trapezoid domain; Third, equiangular octagon domain. In order to get the inner radius of them, we will be based on the classical formular of the Schwarz-Christoffel and improve the proving method of Wieren. We will get a series of inner radius such as parallelogram,sosceles trapezoid and equiangular octagon. Chapter Three, Extreme set of Schwarzian derivative. Because the inner radius of the region is very important to study the analytic function space on the domain. We know that we should estimate the Schwarzian derivative norm when we calculate the inner radius of the domain and it is refer to the extreme set of Schwarzian derivative. In this chapter, we will get the lower bound of inner radius of the equiangular octagon by way of Schwarzian derivative extreme sets.