We find two-sided inequalities for the generalized hypergeometric
function with positive parameters restricted by certain additional conditions.
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. In
the last section of the paper we list some open questions and conjectures.