Inequalities and monotonicity of ratios for generalized hypergeometric function

Abstract. We find two-sided inequalities for the generalized hypergeometric function with positive parameters restricted by certain additional conditions. Our lower bounds are asymptotically precise at x = 0, while the upper bounds are either asymptotically precise or at least agree with q+1Fq((aq+1), (bq);−x) at x = ∞. Inequalities are derived as corollaries of 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 derive the second Thomae relation for 3F2(1) and leads to an integral representations of 4F3(x) in terms of the Appell function F3. In the last section of the paper we list some open questions and conjectures.