### Log-concavity in parameters, two-sided bounds and representations for hypergeometric and related functions

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.

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