Functional inequalities for the quotients of hypergeometric functions

Let F(a, b; c; x) be the Gaussian hypergeometric series and for 0 < r< 1 let[formula]G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen raised recently the following problem: For which a, b ∈ (0, 1) does[formula]hold for all r, s ∈ (0, 1)? They also proved this inequality for a = b = 1/2. The main purpose of this paper is to give an answer to this problem for a + b = 1 and to find a lower bound for the summ(r) + m(s) fora ∈ (0, 2) and b ∈ (0, 2 − a]