关于调和映射、双调和映射和p-调和映射的研究
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摘要
解析函数是复分析中的重要研究对象.作为解析函数的推广,复平面c上的调和映射也越来越得到了人们的关注.1952年,Heinz就利用此类映射来研究单位圆盘上无参最小曲面的Gauss曲率(cf. [1]).而具有里程碑意义是1984年Clunie和Sheil-Small的论文[2].此文表明解析函数的许多经典结果对于调和映射而言仍然成立.作为调和映射的推广,双调和映射来源于许多物理问题,特别是流体力学和弹性问题,故它的研究具有明显的应用特色.作为调和映射和双调和映射的推广,在[3]中,作者定义了p-调和映射,其中p≥1.当p=1(或者p=2)时,即为调和映射(或者双调和映射).
本学位论文主要研究调和映射、双调和映射和p-调和映射的有关性质：首先确定了几种调和映射类和p-调和映射类的极值点和支点；然后讨论了p-调和映射的星形性和凸性；继而研究了双调和映射的Schwarz导数、仿射和线性不变族以及p-调和映射的从属；最后讨论了p-调和映射邻域的存在性.
全文共由六章构成,具体安排如下.
第一章,主要介绍了研究问题的背景和得到的主要结果.
第二章,讨论了调和映射弱从属类的极值点,并把Abu-Muhanna和Hallenbeck在[4]中提出的关于解析函数从属类极值点的弱猜测推广到调和映射情形.所得结果给出了此问题的部分回答.
第三章,给出了双调和映射的Schwarz导数的概念,得到了Schwarz导数解析的一些充分必要条件.同时还给出了双调和映射的仿射和线性不变族的概念,得到了有关Jacob的一些估计.
第四章,主要考虑p-调和映射的从属.首先利用调和映射的分解性质,得到了p-调和映射从属的一个特征；然后考虑了从属p-调和映射积分平均的关系,从而把Schaubroeck在[5]中的相应结果推广到p-调和映射情形；其次确定了p-调和映射从属类闭凸包的两类极值点；最后讨论了p-调和映射从属序列,得到了从属序列的收敛性与该序列对应的偏导数序列收敛性的关系.
第五章,利用系数不等式,确定了两类单叶p-调和映射,并研究了这些p-调和映射的星形性、凸性、极值点和支点,以及p-调和映射邻域的存在性.
第六章,介绍了两类p-调和映射,考虑了它们的性质.首先讨论了p-调和映射的星形性和凸性；然后给出了两个子类的特征；其次确定了这两个子类的极值点；最后讨论了这些p-调和映射的支点和邻域的存在性.
Analytic functions are important objects in the study of complex analysis. As a generalization of analytic functions, planar harmonic mappings attract more and more attention. In 1952, Heinz used these mappings to study the Gauss curvature of nonparametric minimal surfaces (cf. [1]). In the study of harmonic mappings, a landmark paper is [2] in which Clunie and Sheil-Small proved that many of the classical results for analytic functions have analogues in the case of harmonic mappings, and so the study of analytic functions becomes a source of problems for the study of harmonic mappings. As a generaliza-tion of harmonic mappings, biharmonic mappings arise in many physical study, particularly, in fluid dynamics and elasticity problems. As a generalization of harmonic mappings and biharmonic mappings, in [3], the authors introduced the p-harmonic mappings F, where p≥1. Obviously, when p=1 (resp.2), F is harmonic (resp. biharmonic).
The main aim of this thesis is to investigate properties of harmonic, bi-harmonic and p-harmonic mappings. First, we determine the extreme points and support points of several classes of harmonic (resp. p-harmonic) mappings; Then, we discuss the starlikeness and convexity for some p-harmonic mappings; After that, we study the Schwarzian derivatives and the affine and linear invari-ant families of biharmonic mappings, and also the subordination of p-harmonic mappings is discussed; Finally, we discuss the existence of neighborhoods for p-harmonic mappings.
This thesis consists of six chapters. It is arranged as follows.
In Chapter one, we mainly introduce the background of our research and state our main results.
In Chapter two, we discuss the extreme points of weak subordination fami-lies of harmonic mappings and generalize the weak form of the conjecture about extreme points of subordination families of analytic functions to the case of har- monic mappings, which was raised by Abu-Muhanna and Hallenbeck in [4]. Our obtained result is a partial answer to this problem.
In Chapter three, we introduce the Schwarzian derivatives for biharmonic mappings, and obtain several necessary and sufficient conditions for their Schwarzian derivatives to be analytic. Then we prove several estimates related to the Ja-cobian of the functions in the affine and linear invariant families of biharmonic mappings.
In Chapter four, we mainly consider the subordination of p-harmonic map-pings. First, a characterization for p-harmonic mappings to be subordinate is obtained; Second, we investigate the relation of integral means of subordinate p-harmonic mappings. Our result is a generalization of the corresponding one in Schaubroeck [5] to the case of p-harmonic mappings; Third, two classes of ex-treme points of closed convex hulls of the corresponding subordination families are determined; Finally, we discuss the subordinate sequences for p-harmonic mappings and obtain the relation between the convergence of these sequences and the convergence of the corresponding sequences of their partial derivatives.
In Chapter five, by using coefficient inequalities, we introduce two classes of univalent p-harmonic mappings. Then we investigate their starlikeness and convexity, determine the extreme points and support points, and discuss the existence of neighborhoods of these mappings.
In Chapter six, we introduce two classes of p-harmonic mappings and con-sider the properties of these mappings. First, we discuss the starlikeness and convexity of the mappings in these classes; Second, we give the characteriza-tions of two corresponding subclasses; Third, we determine the extreme points of these subclasses; Finally, we consider the determination of the support points of these classes and the existence of neighborhoods of the corresponding map-pings.
本学位论文主要研究调和映射、双调和映射和p-调和映射的有关性质：首先确定了几种调和映射类和p-调和映射类的极值点和支点；然后讨论了p-调和映射的星形性和凸性；继而研究了双调和映射的Schwarz导数、仿射和线性不变族以及p-调和映射的从属；最后讨论了p-调和映射邻域的存在性.
全文共由六章构成,具体安排如下.
第一章,主要介绍了研究问题的背景和得到的主要结果.
第二章,讨论了调和映射弱从属类的极值点,并把Abu-Muhanna和Hallenbeck在[4]中提出的关于解析函数从属类极值点的弱猜测推广到调和映射情形.所得结果给出了此问题的部分回答.
第三章,给出了双调和映射的Schwarz导数的概念,得到了Schwarz导数解析的一些充分必要条件.同时还给出了双调和映射的仿射和线性不变族的概念,得到了有关Jacob的一些估计.
第四章,主要考虑p-调和映射的从属.首先利用调和映射的分解性质,得到了p-调和映射从属的一个特征；然后考虑了从属p-调和映射积分平均的关系,从而把Schaubroeck在[5]中的相应结果推广到p-调和映射情形；其次确定了p-调和映射从属类闭凸包的两类极值点；最后讨论了p-调和映射从属序列,得到了从属序列的收敛性与该序列对应的偏导数序列收敛性的关系.
第五章,利用系数不等式,确定了两类单叶p-调和映射,并研究了这些p-调和映射的星形性、凸性、极值点和支点,以及p-调和映射邻域的存在性.
第六章,介绍了两类p-调和映射,考虑了它们的性质.首先讨论了p-调和映射的星形性和凸性；然后给出了两个子类的特征；其次确定了这两个子类的极值点；最后讨论了这些p-调和映射的支点和邻域的存在性.
Analytic functions are important objects in the study of complex analysis. As a generalization of analytic functions, planar harmonic mappings attract more and more attention. In 1952, Heinz used these mappings to study the Gauss curvature of nonparametric minimal surfaces (cf. [1]). In the study of harmonic mappings, a landmark paper is [2] in which Clunie and Sheil-Small proved that many of the classical results for analytic functions have analogues in the case of harmonic mappings, and so the study of analytic functions becomes a source of problems for the study of harmonic mappings. As a generaliza-tion of harmonic mappings, biharmonic mappings arise in many physical study, particularly, in fluid dynamics and elasticity problems. As a generalization of harmonic mappings and biharmonic mappings, in [3], the authors introduced the p-harmonic mappings F, where p≥1. Obviously, when p=1 (resp.2), F is harmonic (resp. biharmonic).
The main aim of this thesis is to investigate properties of harmonic, bi-harmonic and p-harmonic mappings. First, we determine the extreme points and support points of several classes of harmonic (resp. p-harmonic) mappings; Then, we discuss the starlikeness and convexity for some p-harmonic mappings; After that, we study the Schwarzian derivatives and the affine and linear invari-ant families of biharmonic mappings, and also the subordination of p-harmonic mappings is discussed; Finally, we discuss the existence of neighborhoods for p-harmonic mappings.
This thesis consists of six chapters. It is arranged as follows.
In Chapter one, we mainly introduce the background of our research and state our main results.
In Chapter two, we discuss the extreme points of weak subordination fami-lies of harmonic mappings and generalize the weak form of the conjecture about extreme points of subordination families of analytic functions to the case of har- monic mappings, which was raised by Abu-Muhanna and Hallenbeck in [4]. Our obtained result is a partial answer to this problem.
In Chapter three, we introduce the Schwarzian derivatives for biharmonic mappings, and obtain several necessary and sufficient conditions for their Schwarzian derivatives to be analytic. Then we prove several estimates related to the Ja-cobian of the functions in the affine and linear invariant families of biharmonic mappings.
In Chapter four, we mainly consider the subordination of p-harmonic map-pings. First, a characterization for p-harmonic mappings to be subordinate is obtained; Second, we investigate the relation of integral means of subordinate p-harmonic mappings. Our result is a generalization of the corresponding one in Schaubroeck [5] to the case of p-harmonic mappings; Third, two classes of ex-treme points of closed convex hulls of the corresponding subordination families are determined; Finally, we discuss the subordinate sequences for p-harmonic mappings and obtain the relation between the convergence of these sequences and the convergence of the corresponding sequences of their partial derivatives.
In Chapter five, by using coefficient inequalities, we introduce two classes of univalent p-harmonic mappings. Then we investigate their starlikeness and convexity, determine the extreme points and support points, and discuss the existence of neighborhoods of these mappings.
In Chapter six, we introduce two classes of p-harmonic mappings and con-sider the properties of these mappings. First, we discuss the starlikeness and convexity of the mappings in these classes; Second, we give the characteriza-tions of two corresponding subclasses; Third, we determine the extreme points of these subclasses; Finally, we consider the determination of the support points of these classes and the existence of neighborhoods of the corresponding map-pings.
引文
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