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 F a,b;aqb; 1yr2. m r.s . F a,b;aqb; r2. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen raised recently the following problem: For which a, bg 0, 1.does m r.qm s.F2m ʹrs.hold for all r, sg 0, 1.? They also proved this inequality for asbs1r2. The main purpose of this paper is to give an answer to this problem for aqbs1 and to find a lower bound for the sum m r.qm s. for ag 0, 2. and bg 0, 2yax.