On geometric properties of ratio of two hypergeometric functions

Toshiyuki Sugawa; Li-Mei Wang

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arXiv · 2202.04247

On geometric properties of ratio of two hypergeometric functions

Toshiyuki Sugawa·Li-Mei Wang

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Abstract

R. K\"ustner 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 K\"ustner.

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Toshiyuki Sugawa, Li-Mei Wang. 2022-02-09. On geometric properties of ratio of two hypergeometric functions. https://arxiv.org/abs/2202.04247

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