双解析函数及Schwarz-Pick不等式若干问题的研究
发布时间：2018-11-23 06:49
【摘要】：复变函数是在研究电学、流体力学、空气动力学、理论物理和热力学中发展起来的。数学学科的其他分支和复变函数理论有着紧密的联系。例如初等函数的本质只有在复变函数中方能充分揭示。复变函数的应用非常广泛。在力学、工程力学及物理学的研究中,复变函数起到了关键的作用。本文的主要内容分为四个小节。第一节首先介绍了双解析函数理论的产生背景,发展以及国内外研究现状。其次介绍了Schwarz-Pick不等式以及有界解析零函数n阶导数估计式的发展概况,最后阐述了本论文研究的内容和意义。第二节主要研究了双解析函数的卷绕数定理及其推论。本节首先介绍了一些相关的概念及性质,其次引用了双解析函数的性质,类比解析函数的卷绕数定理,得到了双解析函数的卷绕数定理,最后给出了双解析函数卷绕数定理的三个推论。第三节主要研究了两动点上双曲导数下的Schwarz-Pick不等式。本节首先构造了两个解析映射,且证明了该解析映射符合Schwarz-Pick条件。其次给出了两动点上双曲导数下较强的Schwarz-Pick不等式,最后证明了两动点上双曲导数下更强的Schwarz-Pick不等式。第四节主要研究了有界零函数的n阶导数估计问题。本节首先在已有的n阶导数估计式的基础上,得到了更精确的n阶导数估计式。其次将文中得到的n阶导数估计式与已有结论做了精确性对比,证实了本文估计式的精确性。
[Abstract]:Complex function is developed in the study of electricity, hydrodynamics, aerodynamics, theoretical physics and thermodynamics. Other branches of mathematics are closely related to the theory of complex function. For example, the essence of elementary functions can only be fully revealed in complex functions. Complex function is widely used. Complex function plays a key role in the study of mechanics, engineering mechanics and physics. The main content of this paper is divided into four sections. The first section introduces the background, development and research status of bianalytic function theory. Secondly, the development of Schwarz-Pick inequality and n-order derivative estimator of bounded analytic zero function is introduced. Finally, the content and significance of this paper are expounded. In the second section, we study the winding number theorem of bianalytic functions and its corollary. In this section, we first introduce some related concepts and properties, then we introduce the properties of bianalytic functions, compare the winding number theorems of analytic functions, and obtain the winding number theorems of bianalytic functions. Finally, three corollaries of the twin-analytic function winding number theorem are given. In the third section, we study the Schwarz-Pick inequality under the hyperbolic derivative of the two moving points. In this section, we first construct two analytic mappings and prove that the analytic mappings conform to the Schwarz-Pick condition. Secondly, the stronger Schwarz-Pick inequality under the hyperbolic derivative is given, and finally, the stronger Schwarz-Pick inequality under the hyperbolic derivative is proved. In the fourth section, the problem of n-order derivative estimation of bounded zero functions is studied. In this section, a more accurate estimate of n-order derivative is obtained based on the existing n-order derivative estimators. Secondly, the accuracy of the n-order derivative estimator obtained in this paper is compared with the existing results, which proves the accuracy of the estimator in this paper.
【学位授予单位】：西安建筑科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.5
本文编号：2350669
[Abstract]:Complex function is developed in the study of electricity, hydrodynamics, aerodynamics, theoretical physics and thermodynamics. Other branches of mathematics are closely related to the theory of complex function. For example, the essence of elementary functions can only be fully revealed in complex functions. Complex function is widely used. Complex function plays a key role in the study of mechanics, engineering mechanics and physics. The main content of this paper is divided into four sections. The first section introduces the background, development and research status of bianalytic function theory. Secondly, the development of Schwarz-Pick inequality and n-order derivative estimator of bounded analytic zero function is introduced. Finally, the content and significance of this paper are expounded. In the second section, we study the winding number theorem of bianalytic functions and its corollary. In this section, we first introduce some related concepts and properties, then we introduce the properties of bianalytic functions, compare the winding number theorems of analytic functions, and obtain the winding number theorems of bianalytic functions. Finally, three corollaries of the twin-analytic function winding number theorem are given. In the third section, we study the Schwarz-Pick inequality under the hyperbolic derivative of the two moving points. In this section, we first construct two analytic mappings and prove that the analytic mappings conform to the Schwarz-Pick condition. Secondly, the stronger Schwarz-Pick inequality under the hyperbolic derivative is given, and finally, the stronger Schwarz-Pick inequality under the hyperbolic derivative is proved. In the fourth section, the problem of n-order derivative estimation of bounded zero functions is studied. In this section, a more accurate estimate of n-order derivative is obtained based on the existing n-order derivative estimators. Secondly, the accuracy of the n-order derivative estimator obtained in this paper is compared with the existing results, which proves the accuracy of the estimator in this paper.
【学位授予单位】：西安建筑科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.5
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