  We investigate conditions for logarithmic complete monotonicity of a quotient
of two products of gamma functions, where the argument of each gamma function
has different scaling factor. We give necessary and sufficient conditions in
terms of nonnegativity of some elementary function and 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 parameters under which Meijer's
$G$-function or Fox's $H$-function represents an infinitely divisible
probability distribution on positive half-line. Moreover, we present new
integral equations for both $G$-function and $H$-function. The results of the
paper generalize those due to Ismail (with Bustoz, Muldoon and Grinshpan) and
Alzer who considered previously the case of unit scaling factors.
