  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.
