The Inner Radius Of Univalency For Arc Polygon

The inner radius is an important geometrical features of the universal Teichmüllerspace. It reflects the position of an analytic function and its equivalence classes in theuniversal Teichmüller space, and it is one of the topics in which the complexanalysises are very intrested. It is related to many questions of the geometric functiontheory. The studying of the inner radius is very active all the time.Many scholars suchas Z. Nehari，E. Hille，D. Calvis，L. V. Ahlfors，O. Lehto，M. Lehtinen, F. W.Gehring，L. M. Wieren and so on, come to the inner radius conclusions of a series ofspecial domains such as the unit disk, the triangle, the half plane, the regular n-sidepolygon, the angular domain, and so on.This article concerns the inner radius of univalency for arc polygons and thesimilarities of harmonic function. There are three parts in this thesis.The first part is the preface. In this part, we introduce the basic theory ofquasiconformal mappings, the universal Teichmüller space, the Schwarzian derivativeand the pre-Schwarzian derivative.In part2, we discuss the inner radius of univalency by Schwarzian derivative forarc polygon. According to the Schwarz-Christoffel transformation，when the edge ofthe area are arcs(or line segments) and their images under M bius transformation, theSchwarz-Christoffel formula is determined by the Schwarzian derivative. So we getthe inner radius of univalency by Schwarzian derivative for regular arc n-polygonand extend to the inner radius of univalency by Schwarzian derivative for general arcpolygons.In part3, we discuss the norm of pre-Schwarzian derivative. First, we calculatethe norm of pre-Schwarzian derivative for mappings of unit disk onto itself. Next, weintroduce the properties of the harmonic mappings and the norm of pre-Schwarzianderivative. Last, we calculate the norm of pre-Schwarzian derivative and Schwarzianderivative for convex harmonic mapping.