不同区域上高阶微分从属和微分超从属问题的研究

Differential subordination is a subordination relationship, which is connected with differential inequality and differential equation, while differential superordination is a dual relationship of differential subordination. They are the research hotspot of geometric function theory. In this project, we will investigate the problems of higher-order (strong) differential subordinations and (strong) differential superordinations on different domains in geometric function theory of one complex variable, which include the following four aspects. Firstly, we will establish the theories of nth-order differential subordinations and differential superordinations in the unit disk and study the applications of the above theories on multivalent analytic functions involving some operators. Secondly, we will establish the theories of third-order (strong) differential subordinations and (strong) differential superordinations in the unit disk and discuss the applications of the above theories on multivalent meromorphic functions associated with some operators. Thirdly, we will establish the theories of third-order differential subordinations and differential superordinations in the upper half-plane and consider the applications of the above theories on multivalent analytic functions connected with some operators. Finally, we will establish the theories of third-order (strong) differential subordinations and (strong) differential superordinations in the upper half-plane and point out the applications of the above theories on multivalent harmonic functions defined by using some operators. The main aim of this project is to generalize the classical theories of first-order and second-order (strong) differential subordinations and (strong) differential superordinations in the unit disk to the above higher-order cases on different domains and unify the theories of differential subordinations and differential superordinations in geometric function theory of one complex variable.

微分从属是一种与微分不等式和微分方程有紧密联系的从属关系，而微分超从属是微分从属的一种对偶关系。它们是当前几何函数论学科的研究热点。本项目主要研究单复变几何函数论中不同区域上高阶（强）微分从属与（强）微分超从属问题，具体内容如下：（1）建立单位圆盘内高阶微分从属和微分超从属理论，并讨论该理论在多叶解析函数上关于某些算子的应用；（2）建立单位圆盘内三阶强微分从属和强微分超从属理论，并讨论该理论在多叶亚纯函数上关于某些算子的应用；（3）建立上半平面中三阶微分从属和微分超从属理论，并讨论该理论在多叶解析函数上关于某些算子的应用；（4）建立上半平面中三阶强微分从属和强微分超从属理论，并讨论该理论在多叶调和函数上关于某些算子的应用。本项目的目的是将单位圆盘内一阶和二阶（强）微分从属和（强）微分超从属的经典理论推广到不同区域上的高阶情形，力求扩充和完善单复变几何函数论中的微分从属和微分超从属理论。

从属理论与微分不等式、微分方程有着紧密的联系，是现代数学的重要研究方向之一。本项目主要研究了以下四个方面内容：（1）建立了单位圆盘内高阶微分从属和微分超从属理论，并讨论了该理论关于某些算子的应用；（2）研究了单位圆盘内三阶强微分从属和强微分超从属问题， 并讨论了该理论关于某些算子的应用；（3）建立了上半平面中高阶微分从属和微分超从属理论，并讨论该理论关于某些算子的应用；（4）利用微分从属和超从属理论，研究了某些解析函数的三阶Hankel行列式、三阶Toeplitz行列式以及优化问题。所得结果为多复变函数论中的从属问题、超从属问题及优化问题的研究提供了理论和技术支撑。