平面调和映射中若干问题研究

Planar harmonic mapping is a recent active research topic in the cross-cutting areas of complex analysis and geometry. . In this research project, we aim at studying two problems. Firstly, we shall derive the criteria of univalence for complex-valued harmonic functions defined on simply connected domains in complex plane. Furthermore, we will study the stability properties of harmonic univalent functions by using the properties of univalent harmonic quasiconformal mappings. Secondly, we will consider all the harmonic univalent mappings that mapped the unit disk to a given simply connected domain. By virtue of hypergeometric function and convolution, such results as the forms of representation, boundary behavior properties, convolution properties and subordination properties of harmonic univalent mappings will be studied.. Some innovative thoughts and methods for harmonic univalent mappings will be derived by this research project, which are important for enriching the theory of planar harmonic mappings.

平面调和映射是近年来复分析与几何交叉领域中一个非常活跃的研究热点。. 本项目拟考虑以下问题: 1) 探讨复平面上单连通区域内复值调和函数的单叶性判别准则，再利用单叶调和拟共形映射的相关性质进一步研究调和单叶函数的稳定性；2) 研究通过映射把单位圆盘映入一个给定的单连通区域的调和单叶映射的全体，再结合超几何函数及卷积，探讨它们的表示形式、边界性质、卷积性质及从属性质。. 通过本项目的研究将形成有创新意义的研究调和单叶映射的思路和方法，对丰富平面调和映射理论有着重要意义。

平面调和映射是复分析与几何交叉领域中一个活跃的研究课题。本项目主要研究了以下两个方面的内容： 一是研究了复值调和函数的凸组合及卷积沿着某个方向凸的充分条件，得到了若干关于平面调和映射的单叶性判别准则，并考虑了调和单叶映射的稳定性问题；二是研究了单连通区域内的几类调和单叶映射，得到了它们的表示形式、卷积性质及从属性质。相关研究成果对平面调和映射的单叶性判别等问题具有较重要的意义。