On geometric properties of ratio of two hypergeometric functions

TL;DR

This paper investigates the convexity properties of a ratio of hypergeometric functions, extending previous results by analyzing the order of convexity and addressing open questions from earlier research.

Contribution

It provides new insights into the convexity order of hypergeometric function ratios, partially solving an open problem posed by Küstner in 2002.

Findings

Determined the order of convexity for the ratio function under certain conditions.

Extended the understanding of geometric properties of hypergeometric functions.

Partially resolved an open problem on convexity from previous literature.

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.