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Ukrainian Mathematical Journal

, Volume 64, Issue 4, pp 575–593 | Cite as

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

  • D. S. Skorokhodov
Article

Abstract

We study the following modification of the Landau–Kolmogorov problem: Let k; r ∈ ℕ, 1 ≤ kr − 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.

Keywords

Sharp Estimate Algebraic Polynomial Exact Constant Arbitrary Continuous Function Finite Segment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • D. S. Skorokhodov
    • 1
  1. 1.Dnepropetrovsk National UniversityDnepropetrovskUkraine

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