Precise Estimates for Differences of the Gaussian Hypergeometric Function

By showing certain combinations of the Gaussian hypergeometric functions F a, b; aqb; 1yxc.and F ayd, bqd; aqb; 1yxd.to be monotone on 0, 1. for given a, b, c, dg 0, `., aFb, and c-d, the authors study the problem of comparing these two functions. They find sup d g 0, a.: F a, b; aqb; 1yxc.- F ayd, bqd; aqb; 1yxd.for xg 0, 1.4, thus solving an open problem and a recent conjecture