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 ∈ ℕ, 1 ≤ k ≤ r − 1, and p, q, s ∈ [1,∞]. Also let MM^m, m ∈ 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 ever...
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| Date: | 2012 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2012
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| Series: | Український математичний журнал |
| Subjects: | |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment / D.S. Skorokhodov // Український математичний журнал. — 2012. — Т. 64, № 4. — С. 508-524. — Бібліогр.: 31 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | We study the following modification of the Landau–Kolmogorov problem: Let k; r ∈ ℕ, 1 ≤ k ≤ r − 1, and p, q, s ∈ [1,∞]. Also let MM^m, m ∈ 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 δ > 0, find the exact value of the quantity
We determine the quantity in the case where s = ∞ and m ∈ {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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