The Inner Radius Of Univalency Of Domains

The inner radius of univalency of plane domains is studied in this paper. Some general formulas for the lower bound on inner radius of univalency in the sense of pre-Schwarzian derivative were established. By means of the norm of pre-Schwarzian derivative, three general formulas for the lower bound on inner radius of univalency in the sense of Schwarzian derivative were established. In addition, we can estimate the upper bound on inner radius of univalency for regular polygon by our results.There are three parts in this thesis. The first part is the preface. We introduce the theory of quasiconformal mappings, the theory of Universal Teichmuller Space and the latest developments of them. Furthermore, the problems discussed in this thesis and our main results are introduced.In part 2, we discuss the inner radius of univalency by pre-Schwarzian derivative, which means the problem to find the distance from a point in the Universal Teichmuller Space embedded by pre-Schwarzian derivative to the boundary. Based on any quasidisk, two general formulas for the lower bound on inner radius of univalency in the sense of pre-Schwarzian derivative were established. In addition, By means of the property of regions approaching to given domain, we can get another lower bound on inner radius of univalency. As an application of this formula, we can estimate the upper bound of inner radius of univalency for regular polygon.In part 3, we discuss the relationship between the Universal Teichmuller Space embedded by pre-Schwarzian derivative and the inner radius of univalency by Schwarzian derivative. By means of the norm of pre-Schwarzian derivative, three general formulas for the lower bound of inner radius of univalency in the sense of Schwarzian derivative were established. We can apply the theory of Universal Teichmuller Space to explain their geometric meaning which shows the relationship between the inner radius of univalency by Schwarzian derivative and the norm defined in Universal Teichmuller Space embedded by pre-Schwarzian derivative.