Some sharp inequalities for n-monotone functions

Summary

Some sharp inequalities involving n-monotone functions and their derivatives are obtained. In particular, the following generalization of the Favard-Berwald inequality is established here: \emph{If\/ $f\in C[a,b]$ is non-negative and -f is $(2k)$-monotone, then ${\|f\|}_{C[a,b]}\le \frac{k(k+1)}{b-a}{\|f\|}_{L_1[a,b]}$