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对唯一极值拟共形映射研究的已有结果和最新进展，重点介绍了1998年Bozin V.，Lakic N，Markovic V，和Mateljevic：M．[14]关于唯一极值拟共形映射的研究成果，分析他们的这些极富创新意义的结果，在此基础上我们主要研究了唯一极值拟共形映射的特征刻划，得到了一些重要的和文[14]互不包含的刻划唯一极值拟共形映射的结果。
第三章，四边形的模与本质边界点。在极值拟共形映射理论中，极值映射的最大伸缩商往往是难以计算的，如何解决这个问题也是拟共形映射理论中所要讨论的一个热点。根据拟共形映射下四边形模的拟不变性，利用四边形模之比来逼近它是人们比较容易想到的方法，但关键的问题是四边形模之比的上确界是否等于极值映射的最大伸缩商?在本章中，我们首先利用了文[20]的结果，研究了单位圆周上一
类具有本质边界点的拟对称同胚，证明了它的极值拟共形延拓的最大伸缩商等于四
边形模之比的上确界，改进了文［148］的有关结果．然后，对于抛物区域与双曲区域
上仿射拉伸的边界对应，通过计算，证明上述结论也成立。最后利用退化的四边形
序列，给出了拟对称同胚的极值拟共形延拓的最大伸缩商、四边形模之比的上确界
及拟对称同胚的边界伸缩商三者相等的一个充要条件．
第四章，Schwarz导数与Nehari族．Nehari和Ahlfors对拟共形映射的研究揭示
了Schwarz导数和单叶函数及其拟共形扩张的深刻联系，在本章中，我们首先简单
地介绍了Schwarz导数的研究历史，然后深入分析了Schwarz导数和Nehari族之间
的联系，最后利用微分方程的比较定理和Schwarz导数理论讨论了一个Nehaxi族，
获得了该族的一些重要性质并得到了一系列好的估计．我们不仅发现Nehari族极值
函数和非极值函数的典型性质差别，而且还对SChwarZ导数满足一定增长条件的单
叶函数的像域的拟圆常数作了估计．
第五章，Schwarz导数的极值集．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
我们往往要对Schwarz导数的范数进行估计，这涉及到计算Schwarz导数的极值。
在本章中，我们全面地刻划了Schwarz导数的极值集的分类情况，并对一些特殊的
极值集进行了研究．最后利用Schwarz导数极值集的重要性质部分地解决了矩形和
一类等角六边形的单叶性内径问题．
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 and the theory of Schwarzian derivatives (including Nehari families and the extremal set of Schwarzian derivatives). The main results of this Ph.D. dissertation are briefly introduced in this chapter.
Chapter II, The characteristics of uniquely extremal quasiconformal mappings. In the family of quasiconformal mappings with given boundary correspondence, the extremal mapping must exist, but may be not unique. When is the extremal mapping unique, or what is the characteristic of uniquely extremal quasiconformal mapping is always the key
problem. In this chapter, we first recall the development and the research situation of the theory of uniquely extremal quasiconformal mappings, mainly introduce and analyse the significant results obtained by Bozin V., Lakic N., Markovic V. and Mateljevic M.[14] in 1998. Then we study the characteristics of uniquely extremal quasiconformal mappings, and obtain some criterions of uniquely extremality which are different from the results of [14].
Chapter III, Moduli of quadrilaterals and substantial points. In the theory of quasiconformal mappings, it is often difficult to evaluate the maximal dilatation of the extremal quasiconformal mapping. How to overcome the obstacle is also a hot point. According to the quasi-invariance of the moduli of quadrilaterals under quasiconformal mappings, it is natural to think of approximating the maximal dilatation of the extremal quasiconformal mapping by the ratios of the moduli of quadrilaterals. A key problem is: is it true that the supremum of the ratios of the moduli of quadrilaterals equals the maximal dilatation of the extremal quasiconformal mapping? In this chapter, firstly, we apply a result of [20] to prove that for a class of quasisymmetric homeomorphisms with substantial boundary points, the maximal dilatation of the extremal quasiconformal extension equals the supremum of the ratios of the moduli of quadrilaterals, which improve the result of [148]. Secondly, we prove that the above conclusion is also true for the boundary correspondence of afBne str
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