超几何函数的经典和万有几何性质及应用

The (Gaussian) hypergeometric functions appear in various areas in Mathematics, Physics and Engineering and have proved to be quite useful in many respects. Many of the common mathematical functions can be expressed in terms of hypergeometric functions or their ratios. In complex analysis, they are connected with conformal mappings, quasiconformal theory, differential equations, continued fractions and so on. The use of hypergeometric functions in the confirmation of the Bieberbach conjecture by De Branges has renewed interests to their geometric properties on complex domains...Gaussian hypergeometric functions are the main contents of this project. First of all by determining convergence and range domains of continued fractions with complex elements, we investigate starlikeness and convexity of shifted hypergeometric functions in the unit disk. Then by making use of relations of hypergeometric functions with completely monotone sequences and Pick functions, we consider universal starlikeness and universal convexity of shifted hypergeometric functions and their ratios in the slit domain C\[1,+∞). At last by using the continued fractions and integral forms of ratios of hypergeometric functions, we consider Kustner open problem involving the convolutions of two families of starlike functions.

高斯超几何函数在数学、物理、工程等领域有很多应用，而且几乎所有的初等函数都可以表示成超几何函数或其商的形式。在复分析中，超几何函数与共形映射、拟共形映射、微分方程、连分式等都有密切联系，其在De Branges证明Bieberbach猜想中起的至关作用使研究它在复平面内的几何性质成为热点。. 本项目以超几何函数为研究对象，一方面通过研究复系数连分式的收敛域和值域，探讨移位超几何函数具有星形性、凸性等经典几何性质的条件；另一方面通过考察超几何函数与完全单调序列、Pick函数之间的联系，研究移位超几何函数及其商函数在狭缝区域C\[1,+∞)内具有万有星形性、万有凸性等万有几何性质的条件；最后以超几何函数商的连分式表示和积分表示为工具，对与星形函数族卷积相关的Kustner开问题开展研究。

本项目以超几何函数为研究对象，主要考察它在单位圆盘内的几何性质。一方面给出了移位超几何函数的凸阶数；指出移位零平衡超几何函数在单元圆盘的象域是一个凸区域且位于两条平行于x轴的直线之间。这一结论解决了Ponnusamy和Vuorinen在2001年提出的一个开问题。另一方面考察具有实参数(a,b,c)的移位超几何函数在单位圆盘的凸阶数，首先通过分析超几何函数在z=1点处的渐近性质，给出移位超几何函数的凸阶数为负无穷的充分条件；其次利用斯蒂尔杰斯积分，得到凸阶数为有限实数的充分条件。