Inequalities and monotonicity of ratios for generalized hypergeometric function

Abstract

We find two-sided inequalities for the generalized hypergeometric function of the form ${_{q+1}}F_{q}(-x)$ with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of ${_{q+1}}F_{q}(-x)$ 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 ${_{3}F_{2}}(1)$ and leads to an integral representations of ${_{4}F_{3}}(x)$ in terms of the Appell function $F_3$. In the last section of the paper we list some open questions and conjectures.