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Necessary and sufficient conditions for positive numbers $M_{k_1}, M_{k_2},M_{k_3}, M_{k_4}$, $0 = k_1 < k_2<k_2\leq r-2$, $k_4=r$, to guarantee theexistence of an $r-1$-monotone function defined on the negative half-line andsuch that $\|x^{(k_i)}\| = M_{k_i}$, $i=1,2,3,4$ were found.

Kolmogorov problem on the class of multiply monotone functions