  We find an integral representation for the generalized hypergeometric
function unifying known representations via generalized Stieltjes, Laplace and
cosine Fourier transforms. Using positivity conditions for the weight in this
representation we establish various new facts regarding generalized
hypergeometric functions, including complete monotonicity, log-convexity in
upper parameters, monotonicity of ratios and new proofs of Luke's bounds.
Besides, we derive two-sided inequalities for the Bessel type hypergeometric
functions by using their series representations.

Classical Analysis and ODEs