Teichmuller Space And The Region Of The Inner Radius Of Univalence

This paper is concerned with discussions of the following two quite different but closely related problems:Problemâ… : Suppose that f is holomorphic (or meromorphic) and locally univalent in a plane domain D. What additional conditions on f allow one to conclude that f is univalent?Problemâ…¡: Describe the geometric properties of the Universal Teichmuller Space.The person who first pointed out the relationship of the two problems is Ahlfors. Ahlfors got the relationship by using the method of quasiconformal mapping. Af-ter that, Lehto explained the geometric meaning of the inner radius of univalency by Schwarzian derivative in Universal Teichmuller Space embedded by Schwarzian derivative. We explained the geometric meaning of the inner radius of univalency by pre-Schwarzian derivative in Universal Teichmuller Space embedded by pre-Schwarzian derivative. We can deduce that problemâ… andâ…¡are equivalent when we take account of Schwarzian derivative and pre-Schwarzian derivative.There are five chapters in the thesis. The first chapter is the preface of this the-sis. We introduce the theory of quasiconformal mappings, the theory of Teichmuller 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 boundary of Universal Teichmuller Space embedded by pre-Schwarzian derivative. The model of the Universal Teichmuller Space by the pre-Schwarzian derivative is the union of infinitely many disconnected components: T₁ = {(?)}L. The boundary of T₁ is very complicated. It is proved thatfor anyθ∈[0,2Ï€),(?)L and (?)Lθhave infinitely many common points. In addition, the supremum of the distance from the points of one component to the center of another component is solved. In chapter 3, we discuss the inner radius of univalency by pre-Schwarzian deriv-ative, that is the problem to find the distance from a point in the Universal Teich-muller Space embedded by pre-Schwarzian derivative to the boundary. Some general formulas for the lower bound of inner radius are established. The known results on inner radius of univalency by pre-Schwarzian derivative become our corollaries. Furthermore, as their applications, the lower bounds of inner radiuses for angular domains and strongly starlike domains are obtained.In chapter 4, we discuss the relationship between the Universal Teichmuller Space embedded by pre-Schwarzian derivative and the inner radius of univalency by Schwarzian derivative. We get a lower bound of inner radius of univalency by Schwarzian derivative by means of the norm of pre-Schwarzian derivative. Further-more, we apply the theory of Universal Teichmuller Space to explain its geomet-ric meaning which shows the relationship between the inner radius of domains by Schwarzian derivative and the norm defined in Universal Teichmuller Space embed-ded by pre-Schwarzian derivative.In chapter five, we discuss the problem about pre-Schwarzian derivative and quasiconformal extension. This problem also relates to the Universal Teichmuller Space embedded by pre-Schwarzian derivative. We find some connections between the complex dilatations of the quasiconformal extensions and the norms of the pre-Schwarzian derivatives. Furthermore, we find another proof for the lower bound of the inner radius of univalency for angular domains by constructing an explicit quasiconformal extension of a class of holomorphic functions.