Carathéodory函数的研究及应用

Carathéodory函数(也称为正实部函数)不仅是几何函数理论的重要研究对象，而且它还与插值问题、正交多项式、连分数、概率论、随机过程等重要数学问题联系密切，在信号分析等许多学科当中发挥着重要作用。本项目拟从正实部函数的系数参数化以及它与Löwner-Kufarev (L-K)微分方程之间的内在联系两方面研究的基础上，对单叶函数和单叶调和函数中的几个极值问题展开探讨。其具体内容包括：(1)运用Schur参数和连分数理论将正实部函数的系数进行参数化，并且应用参数化的方法解决与Zalcman泛函与相邻系数模差泛函相关的两个单叶函数极值问题。(2)通过对L-K微分方程与正实部函数之间的关系研究，找到拥有具体形式的更广单叶函数子族。(3)通过L-K微分方程与Φ-型函数的研究，寻找一类重要调和函数的构造方法。本项目的研究可以为Zalcman猜想和调和Bieberbach猜想的研究提供新思路。

Carathéodory function is not only a central topic in Geometric Function Theory, but also closely related to the significant mathematical problems such as interpolation problems, orthogonal polynomials, continued fractions, probability theory and stochastic processes. It is playing an important role in signal analysis and many other subjects. In this program, we will study the parametric coefficients of Carathéodory functions, and the relationship between Carathéodory functions and the Löwner-Kufarev (L-K) differential equation. Based on these study, we aim at finding solutions for several extremal problems in Geometric Function Theory and planar harmonic mappings. The main research contents are as follows: .(1) The coefficients of Carathéodory functions will be parameterized with the application of the iteration method and continued fraction theory. By using the parametric coefficients, we study two extremal problems on Geometric Function Theory which is related to the Zalcman functional, and the difference of mududi successive coefficients functional. .(2) Do research on the inherent relationship between the Carathéodory function and the L-K differential equation. We are to find a larger subclass of univalent functions..(3) We study the Φ-like funtctions and the L-K differential equation to construct a specific subclass of harmonic functions..This program will give new ideas for the study of Zalcman conjecture in Geometric Function Theory, and the analogy Bieberbach conjecture in planar harmonic mappings.