Inequalities for Zero-Balanced Gaussian hypergeometric function

Abstract

In this paper, we consider the monotonicity of certain combinations of the Gaussian hypergeometric functions $F(a-1,b;a+b;1-x^c)$ and $F(a-1-\delta ,b+\delta ;a+b;1-x^d)$ on $(0,1)$ for $\delta \in (a-1,0)$, and study the problem of comparing these two functions, thus get the largest value ${\delta }_1={\delta }_1(a,c,d)$ such that the inequality $F(a-1,b;a+b;1-x^c)<F(a-1-\delta ,b+\delta ;a+b;1-x^d)$ holds for all $x\in (0,1)$.

10 pages