Investigations On The Inner Radius Of Univalence For Some Domains

The present dissertation is concerned with some problems in the inner radius of univalence of some parallelogram and some inequiangular hexagon.The research on the inner radius of univalence attracts many people's attention. Galvis, Lehto, Lehtinen, Miller-VanWieren, Ahlfors, Gehring, Nehari and Hille obtained many important results. The inner radius of univalence for some particular domains such as triangles, regular polygons and angle sectors has been obtained.There are four parts in this paper. Chapter I, Introduction. We introduce the basic theory of quasiconformal mappings of the development and the research situation of the theory of quasiconformal mappings, the theory of Schwarz derivatives and the inner radius of univalence. The main results of this paper are briefly introduced in this chapter. Chapter II, The inner radius of univalence for some parallelogram. Using the methods developed by Wieren, we obtain the innerradius of univalence of some classes of parallelogram R_α with some given α. In the proff, we use the Mathematica software package. In chapter III, we discuss the inner radius of univalence for some inequiangular hexagon H_α whoseangularities form the sequence αββαββ and sides form the sequence baabaa (α= kπ,a,b depend on k). Using the methods developed by Wieren, we obtain that for some given α, σ(H_α) = 2k~2. Chapter IV, The inner radius of univalenceof a domain plays an important role in the study of the univalence and other properties of analytic functions on the domain. In order to obtain the inner radius of univalence of a domain, we must make estimations to the norms of Schwarz derivatives, which involve computing the extremal values of Schwarz derivatives. In this chapter, we partially solve the problem of inner radius of univalence...