If exp {−tH}, exp {−tK}, are self-adjoint, positivity preserving, contraction semigroups on a Hilbert space ℋ=L2(X;dμ) we write $$e^{ - tH} \succcurlyeq e^{ - tK} \succcurlyeq 0$$ (*) whenever exp {−tH}-exp {−tK} is positivity preserving for allt≧0 and then we characterize the class of positive functions for which (*) always implies $$e^{ - tf(H)} \succcurlyeq e^{ - tf(K)} \succcurlyeq 0.$$ This class consists of thef∈C∞(0, ∞) with $$( - 1)^n f^{(n + 1)} (x) \geqq 0,x \in (0,\infty ),n = 0,1,2, \ldots .$$ In particular it contains the class of monotone operator functions. Furthermore if exp {−tH} isLp(X;dμ) contractive for allp∈[1, ∞] and allt>0 (or, equivalently, forp=∞ andt>0) then exp {−tf(H)} has the same property. Various applications to monotonicity properties of Green's functions are given.