两个超几何函数之比的几何性质
On geometric properties of ratio of two hypergeometric functions
摘要：
R. Küstner在他2002年的论文中证明了函数$w_{a,b,c}(z)=$ $F(a+1,b;c;z)/F(a,b;c;z)$将单位圆盘$|z|<1$映射到一个在虚轴方向上的凸域上，在一定条件下，实参数$a,b,c.$在这里$F(a,b;c;z)$代表高斯超几何函数。本文研究了$w_{a,b,c}.$的凸性阶，特别地，我们部分地解决了Küstner的上述文章中提出的问题.
英文摘要：
R. Küstner proved in his 2002 paper that the function $w_{a,b,c}(z)=$ $F(a+1,b;c;z)/F(a,b;c;z)$ maps the unit disk $|z|<1$ onto a domain convex in the direction of the imaginary axis under some condition on the real parameters $a,b,c.$ Here $F(a,b;c;z)$ stands for the Gaussian hypergeometric function. In this paper, we study the order of convexity of $w_{a,b,c}.$ In particular, we partially solve the problem raised by the afore-mentioned paper by Küstner.
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