Completely monotonic gamma ratio and infinitely divisible H-function of Fox

We investigate conditions for the logarithmic complete
monotonicity of a quotient of two products of gamma functions, where
the argument of each gamma function has a different scaling factor. We
give necessary and sufficient conditions in terms of non-negativity of
some elementary functions and some more practical sufficient
conditions in terms of parameters. Further, we study the representing
measure in Bernstein’s theorem for both equal and non-equal scaling
factors. This leads to conditions on the parameters under which Meijer’s
G-function or Fox’s H-function represents an infinitely divisible
probability distribution on the positive half-line.