On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment

Abstract

We study the following modification of the Landau-Kolmogorov problem: Let $k, r \in \mathbb{N}, \quad 1 \leq k \leq r -1$ and $p, q, s \in [1, \infty]$. Also let $MM^m,\; m \in \mathbb{N}$, be the class of nonnegative functions defined on the segment $[0,1]$ whose derivatives of orders $1, 2,... , m$ are nonnegative almost everywhere on $[0,1]$. For every $\delta > 0$, find the exact value of the quantity $$w^{k, r}_{p, q, s}(\delta; MM^m) := \sup \left\{ ||x^{(k)}||_q : \; x \in MM^m,\; ||x||_p \leq \delta, \;\; ||x^{(r)}||_s \leq 1\right\}$$ We determine the quantity $w^{k, r}_{p, q, s}(\delta; MM^m)$ in the case where $s = \infty$ and $m \in \{r,\; r — 1,\; r — 2\}$. In addition, we consider certain generalizations of the above-stated modification of the Landau-Kolmogorov problem.