Inequalities for the generalized hypergeometric function are surprisingly rare in the literature. In the talk
we consider several interrelated classes of such inequalities. First we address the following question: under
what conditions on parameters the generalized hypergeometric function is log-concave as a function of
simultaneous shift of its several parameters? Similar question may be asked about log-convexity but the
answer iis then more accessible due to additivity of log-convexity. We give a complete answer for shifts in
one parameter and partial answer for shifts in two parameters for a generic series in product ratios of rising
factorials and gamma functions. Next we discuss generalized Stieltjes transforms of a non-negative measures
summarizing some of their known and new properties. We introduce the notion of exact Stieltjes order and
nd this order for generalized hypergeometric function. Finally, we show how generalized Stieltjes transform
representation of generalized hypergeometric function leads to monotony of their ratios, asymptotically
precise two-sided bounds improving and extending previous results due to Y.Luke and B.C.Carlson and to
log-convexity in parameters for negative values of the argument when additivity cannot be applied. We
also formulate several conjectures, some of analytic and some of combinatorial nature.
** This is a joint work with: S. M. Sitnik and S. Kalmykov.