  In this paper we give necessary and sufficient conditions for the system of
positive numbers $ M_{k_1}, M_{k_2},..., M_{k_{d}},$ $0\leq k_1<...<k_{d}
{\leq} r$, to guarantee the existence of an $r$-monotone function defined on
the negative half-line $\RR_-$ and such that $\|x^{(k_i)}\|_{\infty}=M_{k_i},
i=1,2,...,d$.
