单叶函数、调和测度与渐近共形延拓

发布时间：2018-06-17 16:02

本文选题：拟共形映射 + Loewner链　；参考：《南京理工大学》2017年硕士论文

【摘要】：本文主要研究单位圆内(或单位圆外)单叶函数的拟共形延拓问题以及渐进共形曲线的调和测度刻画问题。首先研究Loewner链理论,并利用Loewner链刻画了单叶函数能渐近共形延拓的充分性条件。其次,用调和测度刻画了渐近共形曲线。最后研究了形如f(z)= z + ω(z)(或f(z)=z + ω(1/z)的单叶函数渐近共形延拓问题。论文共分四章,第一章我们将简要介绍拟共形映射相关概念及其发展情况,并叙述本文研究的问题和所得结论。第二章,我们利用Loewner链理论研究单叶函数。Becker利用Lowner链给出了单位圆盘上单叶函数可以拟共形延拓的一个充分条件。类似于Becker的结果,我们利用Loewner链给出了单位圆盘上单叶函数可以渐近共形延拓的一个充分条件。进一步,我们将所得结果运用到一些特殊单叶函数族中(例如凸函数,近凸函数),并给出了这些子类能渐近共形延拓的充分条件。第三章,我们用调和测度去刻画渐近共形曲线。利用调和测度,Hag等人给出了拟圆周的一种刻画。类似于Hag的结果,本文给出了渐近共形曲线的调和测度刻画。在第四章中我们研究了形如f(z)=z+ω(z)(或f(z)=z+ω(1/z))的单叶函数渐近共形延拓问题,并将其进行推广。
[Abstract]:In this paper, we mainly study the quasi-conformal continuation problem of univalent functions in and out of unit circles and the harmonic measure characterization of asymptotically conformal curves. Firstly, the Loewner chain theory is studied, and the sufficient conditions for asymptotically conformal continuation of univalent functions are described by using Loewner chain. Secondly, asymptotically conformal curves are characterized by harmonic measures. Finally, the asymptotically conformal continuation problem of univalent functions such as fnzn = z 蠅 zn (or f(z)=z 蠅 n 1 / z) is studied. This paper is divided into four chapters. In the first chapter, we will briefly introduce the concept and development of quasi-conformal mapping, and describe the problems and conclusions of this paper. In chapter 2, we use Loewner chain theory to study univalent function. Becker gives a sufficient condition that univalent function can be quasi-conformal extension by Lowner chain. Similar to Becker's results, we give a sufficient condition for univalent functions on a unit disk to be asymptotically conformal by using Loewner chains. Furthermore, we apply the results to some special families of univalent functions (such as convex functions, nearly convex functions), and give sufficient conditions for these subclasses to be asymptotically conformal. In chapter 3, we use harmonic measures to characterize asymptotically conformal curves. By means of harmonic measure Hag et al., we give a characterization of quasi-circumference. Similar to Hag's results, the harmonic measures of asymptotically conformal curves are given in this paper. In chapter 4, we study the asymptotically conformal continuation problem of univalent functions such as f(z)=z 蠅 zn (or f(z)=z 蠅 n 1 / z), and generalize it.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.51

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