Some sharp inequalities for \(n\)-monotone functions (Q2568506)

Let \(n\in \mathbb N\) and \(f\:[a,b]\rightarrow\mathbb R\). Recall that \(f\) is said to be \(n\)-monotone on \([a,b]\) if, for all choices of distinct points \(x_0,\dots,x_n\) in \([a,b]\), \(f[x_0,\dots,x_n]\geq 0\), where \(f[x_0,\dots,x_n]\) is the \(n\)-th order divided difference of \(f\) at \(x_0,\dots,x_n\). The aim of the paper is to investigate inequalities involving \(n\)-monotone functions. To be more specific, we mention two of them: (i) Let \(f\in C[a,b]\), \(f\geq 0\), and let \((-f)\) be \((2k)\)-monotone on \([a,b]\). Then \[ \|f\|_{C[a,b]} \leq \frac{k(k+1)}{b-a} \|f\|_{L_1 [a,b]}. \] (ii) Let \(f\in C^{n-1}[a,b]\) be \(n\)-monotone on \([a,b]\) and let \(a< c < d<b\). Then \[ \|f^{(n-1)}\|_{C[c,d]} \leq K \|f\|_{C[a,b]} \] and \[ \|f^{(n-1)}\|_{L_1[c,d]} \leq K \|f\|_{L_1 [a,b]}, \] where \(K = (n-1)!\,2^{2n-3} (\max \{(c-a)^{-1}, (b-d)^{-1}\})^{n-1}\).