导　　师： 陈纪修

学科专业： 070101

授予学位： 博士

作　　者： ;

机构地区： 复旦大学

摘　　要： 本文的主要目的在于研究拟共形映射极值问题及与之相关的Schwarz导数理论。拟共形映射是复变函数论中共形映射/(或称保角变换/)的拓广。从1928年Gr/(?/)lzsch提出至今已有七十多年的历史，在这几十年中，伴随着对它的研究的进一步深入，拟共形映射理论已经渗透到数学、物理、科技和工程等各个领域，对其它学科的研究提供了有力的研究工具。 拟共形映射极值理论主要讨论给定边界对应的拟共形映射族中极值映射的存在性、唯一性、及极值映射的性质与特征刻划等问题。其中唯一极值拟共形映射的特征刻划以及相关的一些问题一直是研究的热点和难点。本文的第二章及第三章对这些问题进行了深入的研究，得到了一系列的结果。 Schwarz导数在判定共形映射能否拟共形延拓、估计区域的单叶性内径以及探讨一些解析函数族的性质方面有非常重要的作用，对这些热点问题的研究将对拟共形映射理论的发展起着积极的作用。在第四章和第五章中，对解析函数的Schwarz导数和Nehari族以及Schwarz导数的极值集作了深入细致的研究，并且利用所得到的结果研究了矩形、等角六边形的单叶性内径问题。 第一章，绪论。在这一章中，我们简单介绍了拟共形映射的基本理论，拟共形映射极值问题、Schwarz导数理论/(包括有关的Nehari族与Schwarz导数的极值集/)的发展历史与研究现状，并对论文的主要结果给以简单介绍。 第二章，唯一极值拟共形映射的特征刻划。在给定边界值的拟共形扩张中，一定存在极值拟共形映射，但极值映射不一定是唯一的。因此对于给定的边界值，什么时候存在唯一极值拟共形扩张，也就是唯一极值拟共形映射的特征刻划一直是一个热点、难点问题。在这一章中，我们首先简要回顾了� The present Ph.D. dissertation is concerned with the extremal problems in the theory of quasiconformal mappings and the related topics: quasiconformal extensions and Schwarzian derivatives. Quasiconformal mapping, which was posed by Grotzsch in 1928, is the generalization of conformal mapping in the theory of complex analysis. During the several decades, with the development of its theory, it has been widely spread into many research fields such as physics, science and technology, engineering, and other branches in mathematics, and provide a powerful tool for the study and research in these fields. The theory of extremal quasiconformal mappings is mainly concerned with the problems of existence and uniqueness of extremal quasiconformal mappings with given boundary correspondence and of the properties and characteristics of extremal quasiconformal mappings. Among which the problem of the characteristics of uniquely extremal quasiconformal mappings is the most difficult one and is most widely concerned. We discuss these problems in the second and third chapters of this paper, and obtain a series of deep results. The theory of Schwarzian derivatives has great significance in determining whether a conformal mapping has quasiconformal extensions, in estimating the inner radius of uni-valence of a domain and in discussing the properties of some conformal mapping families. The study of these key problems will be very important to the development of the theory of quasiconformal mappings. In the fourth and fifth chapters of this paper, we discuss the Schwarzian derivatives of analytic functions, the Nehari families and the extremal set of Schwarzian derivatives, and apply the obtained results to determine the inner radius of univalence of rectangles and hexagons with equal angles. Chapter I, Preface. This chapter is devoted to the exposition of the basic theory of quasiconformal mappings, of the development and the research situation of the theory of extremal quasiconformal mappings

关 键 词： 拟共形映射 唯一极值拟共形映射 族 导数 单叶性内径

领　　域： [理学] [理学]