Completely Monotonic Rational Functions and Hall′s Marriage Theorem

It is shown that the rational function given by ƒ(x) = ∏ni = 1((x+ zi)/(x+pi)) is completely monotonic if 0 ≤ p1 ≤ ··· ≤ pn, 0 ≤ z1 ≤ ··· ≤ zn and for each k = 1, ..., n, ∑k1zi ≥ ∑k1pi.