关于区域单叶性内径的若干研究/Investigations on the inner radius of univalence for some domains

2019-02-26 14:21:47

inner radius 内径 单叶 univalence



关于单叶性内径的研究一直十分活跃,Calvis、Lehto、Lehtinen、Wieren、 Ahlfors、Gehring、Nehari、Hille等学者得到了一系列的结果。对三角形、正多边形、角形区域、双曲线围成的区域的单叶性内径已经得到了精确的数值。
本文主要研究了平行四边形和不等角六边形的单叶性内径问题。全文共分为四个部分。第一章,绪论。在这一章中,我们简单介绍了拟共形映照的基本理论,回顾了拟共形映照及Schwarz导数理论的发展及区域单叶性内径的研究现状,并简要的介绍了作者的主要工作。第二章,几类平行四边形的单叶性内径。对平行四边形的单叶性内径,我们从经典的Schwarz-Christoffel公式出发,利用Wieren的证明方法,并借助于Mathematica软件包,得到了与一些给定的 值相对应的几类平行四边形 的单叶性内径 。推广了Wieren对矩形单叶性内径研究的结果。第三章,几类六边形的单叶性内径。我们同样利用Wieren的证明方法,并借助于Mathematica软件包,对边长序列为 ,角序列为 的不等角六边形 的单叶性内径进行了研究,得到了与一些给定的 值相对应的几类不等角六边形 的单叶性内径 。推广了Wieren的边长序列为 的等角六边形的单叶性内径的结果。第四章,用Schwarz导数极值集的性质解区域的单叶性内径。我们知道一个区域的单叶性内径对研究该区域上解析函数的单叶性和其他性质具有很重要的意义,而计算区域的单叶性内径时我们要对Schwarz导数的范数进行估计,这涉及到计算Schwarz 导数的极值。在这一章中我们利用Schwarz 导数极值集的重要性质部分的解决了几类平行四边形的单叶性内径。



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 inner radius of univalence of some classes of parallelogram with some given . In the proff, we use the Mathematica software package. In chapter III, we discuss the inner radius of univalence for some inequiangular hexagon whose angularities form the sequence and sides form the sequence ( depend on ). Using the methods developed by Wieren, we obtain that for some given , . Chapter IV, The inner radius of univalence of 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 for some parallelogram, applying the properties of extremal sets of Schwarz derivatives.