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

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Abstract

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

$$ \omega_{p,q,s}^{k,r}\left( {\delta; M{M^m}} \right): = \sup \left\{ {{{\left\| {{x^{(k)}}} \right\|}_q}:x \in M{M^m},{{\left\| x \right\|}_p} \leqslant \delta, {{\left\| {{x^{(k)}}} \right\|}_s} \leqslant 1} \right\}. $$

We determine the quantity \( \omega_{p,q,s}^{k,r}\left( {\delta; M{M^m}} \right) \) 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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Authors and Affiliations

  1. Dnepropetrovsk National University, Dnepropetrovsk, Ukraine

    D. S. Skorokhodov

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  1. D. S. Skorokhodov
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 4, pp. 508–524, April, 2012.

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Skorokhodov, D.S. On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment. Ukr Math J 64, 575–593 (2012). https://doi.org/10.1007/s11253-012-0665-6

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  • DOI: https://doi.org/10.1007/s11253-012-0665-6

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