On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment
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,......
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| Date: | 2012 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2012
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2592 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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. |
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