【作者】 杨宗信；

【导师】 陈纪修；

【作者基本信息】 复旦大学 ， 基础数学， 2004， 博士

【摘要】 本文主要研究拟共形映照及与之相关的Schwarz导数及拟共形延拓问题。 拟共形映照理论是复变函数论中共形映照理论的拓展．从1928年Gr(?)tzsch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逐步深入，拟共形映照理论已经渗透到数学其它分支、物理学和工程技术等各个领域，为其它学科的发展提供了有力的研究工具。 本文共分五章： 第一章，绪论。在这一章中，我们简单介绍拟共形映照的基本理论，回顾拟共形映照及Schwarz导数理论(主要包括Nehari函数族的分析与几何性质)的发展历史与研究现状，并简要地介绍作者的主要工作。 第二章，Nehari函数族的偏差定理与拟共形延拓。我们称满足Nehari单叶性判据的解析函数全体所组成的集合为Nehari族。Nehari关于函数单叶性及Ahlfors和Weill关于拟共形延拓的研究揭示了Schwarz导数与单叶函数及其拟共形延拓的深刻联系。在本章中，我们利用Schwarz导数与二阶线性微分方程的关系，运用微分方程解的比较定理，讨论了一类Nehari函数的偏差性质与拟共形延拓，获得了这一类Nehari函数的几个重要偏差性质，推广了Chuaqui，Gehring，Osgood和Pommerenke等人的若干结果。我们还构造了这类函数的拟共形延拓的具体表达式，推广了Ahlfors和Weill的结果。 第三章，Schwarz导数与John区域。John区域可以看成是满足拟圆单边条件的区域，若有界区域Ω与Ω~*=(?)\Ω均为John区域，则Ω是拟圆。在这一章中，我们研究了Schwarz导数满足(1-｜z｜~2)｜S_f(z)｜＜4这一Nehari函数子族的一些特殊的偏差性质。在此基础上，我们研究了这类函数与John区域的关系，并讨论了一个与Schwarz导数、对数导数及John区域密切相关的函数的偏差定理，给出了Ω=f(D)为John区域的一个充分条件。 第四章，对数导数与拟共形延拓。根据对数导数的增长性与函数单叶性的关系，我们研究了一类单叶函数的偏差性质及其拟共形延拓，并给出了拟共形延拓的具体表达式， 第五章，拟Fu比s群的收敛指数.根据呼上的双曲距离在拟共形变换下的拟不变性，我们研究了K一拟共形群的收敛指数，给出了K一拟共形抛物循环Fuch。群的收敛指数的估计， 第六章，极值拟共形映照的极值集，探讨极值拟共形映照与无限小极值B eltr二i系数的极值集.设f(:)是单位圆到单位圆的极值拟共形映照，以户为复特征，对于厂‘的复特征乒，存在它的极值集X同的正测度紧子集户，使 1 In士-;~，二~，，~，~，~一，-，任Q:(D)J坛}沪ldudy(，!;}}二一厂汇。、动，0，\JJ妙/则在川z)的等价类中，必存在极值复特征州:)，使X回属于X回一E，其中乒(叫二脚一、，E二了一‘(自.无限小极值Beltrami系数也有类似结果.更多还原

【Abstract】 The present Ph.D. dissertation is concerned with the extremal problems in the theory of quasiconformal mappings and the related topics: Schwarzian derivative, quasiconformal extension and John domain.Chapter I: Preface. This chapter is devoted to the exposition of the basic theory of quasiconformal mappings, of the development and the reseach situation of the theory of quasiconformal mappings and the theory of Schwarzian derivatives, including the analytic and geometric properties of functions in the Nehari class. The main results of this Ph.D. dissertation are briefly introduced in this chapter.Chapter II: On the distortion theorems and quasiconformal extensions of the Nehari class. Denote the family of analytic functions satisfying Nehari’s univalence criteria by Nehari class. The researches on the theory of univalency criteria of analytic functions by Nehari and the theory of the quasiconformal extensions by Ahlfors and Weill revealed the deep connection between the Schwarzian derivative and the quasiconformal extension of univalent functions. In this chapter, we first analyse the relationship between Schwarzian derivative and the second order linear ordinary differential equation, and then by using the comparison theorems of ordinary differential equation, we study the distortion properties and quasiconformal extensions of a class of Nehari functions. In the last section, we construct an explicit quasiconformal extension of this class of Nehari function. Our work extended some results obtained by Chuaqui and Osgood, Gehring and Pommerenke, Ahlfors and Weill.Chapter III: John disks and the Schwarzian derivative. John disks can be thought as " one-sided quasidisks ", a Jordan domain Cl C C is a quasidisk if and only if and * = C \ are John disks. This chapter is concerned with functions in a subclass of the Nehari class whose Schwarzian derivatives satisfy (1 - |z|2)|Sf(z)| < 4. We discuss the connection between those functions and John disks. We also prove some results concerning a new function which is related to the Schwarzian derivative, logarithmic derivative and John disk. Then we give a new sufficient condition on for = f(D) to be a John disk.Chapter IV: Logarithmic derivative and quasiconformal extension. Using the relationship between the logarithmic derivative and the univalency of a function, we discuss the distortion properties and quasiconformal extension of a class of univalent functions, and then an explicit quasiconformal extension of this class is obtained.Chapter V: The exponent of convergence of quasiconformal Fuchsian groups. Using the quasi-invariant property of the hyperbolic distance under a quasiconformal mapping, we discuss the exponent of convergence of quasiconformal Fuchsian groups, and then we give an estimate related to a conjecture which is posed by Bonfert-Taylor and Taylor.Chapter VI: On the extremal set of extremal quasiconformal mapping. Let f(z) be an extremal quasiconformal mapping of the unit disk D onto itsef, X[p] - [z z G D, \n(z)\ - Halloo}, A = /*/~1- ^ there exists a compact set E C X[p] with mesE > 0, satisfyingthen there exists an extremal Beltrami coefficients v(z] ~ M(Z), which satisfies X[v\ C X[] -E, where E = f-1(E).We also obtain a similar result on infinitesimally extremal Beltrami coefficients. At the end, we obtain a sufficient condition for (Z) to be uniquely extremal.更多还原