本文選題：擬共形映射 + Loewner鏈��； 參考：《南京理工大學》2017年碩士論文

【摘要】：本文主要研究單位圓內(nèi)(或單位圓外)單葉函數(shù)的擬共形延拓問題以及漸進共形曲線的調(diào)和測度刻畫問題。首先研究Loewner鏈理論,并利用Loewner鏈刻畫了單葉函數(shù)能漸近共形延拓的充分性條件。其次,用調(diào)和測度刻畫了漸近共形曲線。最后研究了形如f(z)= z + ω(z)(或f(z)=z + ω(1/z)的單葉函數(shù)漸近共形延拓問題。論文共分四章,第一章我們將簡要介紹擬共形映射相關概念及其發(fā)展情況,并敘述本文研究的問題和所得結論。第二章,我們利用Loewner鏈理論研究單葉函數(shù)。Becker利用Lowner鏈給出了單位圓盤上單葉函數(shù)可以擬共形延拓的一個充分條件。類似于Becker的結果,我們利用Loewner鏈給出了單位圓盤上單葉函數(shù)可以漸近共形延拓的一個充分條件。進一步,我們將所得結果運用到一些特殊單葉函數(shù)族中(例如凸函數(shù),近凸函數(shù)),并給出了這些子類能漸近共形延拓的充分條件。第三章,我們用調(diào)和測度去刻畫漸近共形曲線。利用調(diào)和測度,Hag等人給出了擬圓周的一種刻畫。類似于Hag的結果,本文給出了漸近共形曲線的調(diào)和測度刻畫。在第四章中我們研究了形如f(z)=z+ω(z)(或f(z)=z+ω(1/z))的單葉函數(shù)漸近共形延拓問題,并將其進行推廣。
[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.
【學位授予單位】：南京理工大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O174.51

【參考文獻】

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本文編號：2031615