Inequalities and monotonicity of ratios for generalized hypergeometric function

We find two-sided inequalities for the generalized hypergeometric function of the form q+1Fq(−x)q+1Fq(−x) with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of q+1Fq(−x)q+1Fq(−x) at the endpoints of positive semi-axis and are asymptotically precise at one of the endpoints. The inequalities are derived from a theorem asserting the monotony of the quotient of two generalized hypergeometric functions with shifted parameters. The proofs hinge on a generalized Stieltjes representation of the generalized hypergeometric function. This representation also provides yet another method to deduce the second Thomae relation for 3F2(1)3F2(1) and leads to an integral representations of 4F3(x)4F3(x) in terms of the Appell function F3F3. In the last section of the paper we list some open questions and conjectures.