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


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.


Sharp Estimate Algebraic Polynomial Exact Constant Arbitrary Continuous Function Finite Segment 
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  1. 1.
    V. F. Babenko and Yu. V. Babenko, “Kolmogorov inequalities for multiply monotone functions defined on a half-line,” East J. Approxim., 11, No. 2, 169–186 (2005).MathSciNetzbMATHGoogle Scholar
  2. 2.
    V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Additive inequalities for intermediate derivatives of differentiable mappings of Banach spaces,” Math. Notes, 63, No. 3, 332–342 (1998).MathSciNetCrossRefGoogle Scholar
  3. 3.
    V. F. Babenko, N. P. Korneichuk, V. A. Kofanov, and S. A. Pichugov, Inequalities for Derivatives and Their Applications [in Russian], Naukova Dumka, Kiev (2003).Google Scholar
  4. 4.
    V. F. Babenko and T. M. Rassias, “On exact inequalities of Hardy–Littlewood–Pólya type,” J. Math. Anal. Appl., 245, 570–593 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Yu. V. Babenko, “Pointwise inequalities of Landau–Kolmogorov type for functions defined on a finite segment,” Ukr. Math. J., 53, No. 2, 270–275 (2001).MathSciNetCrossRefGoogle Scholar
  6. 6.
    B. Bojanov and N. Naidenov, “An extension of the Landau–Kolmogorov inequality. Solution of a problem of Erdös,” J. D’Anal. Math., 78, 263–280 (1999).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    B. Bojanov and N. Naidenov, “Examples of Landau–Kolmogorov inequality in integral norms on a finite interval,” J. Approxim. Theory, 117, 55–73 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, Berlin (1995).zbMATHCrossRefGoogle Scholar
  9. 9.
    V. I. Burenkov, “Exact constants in inequalities for norms of intermediate derivatives on a finite interval. I,” Tr. Mat. Inst. Steklov., 156, 22–29 (1980).MathSciNetzbMATHGoogle Scholar
  10. 10.
    V. I. Burenkov, “Exact constants in inequalities for norms of intermediate derivatives on a finite interval. II,” Tr. Mat. Inst. Steklov., 173, 38–49 (1986).MathSciNetzbMATHGoogle Scholar
  11. 11.
    V. I. Burenkov and V. A. Gusakov, “On sharp constants in inequalities for the modulus of a derivative,” Tr. Mat. Inst. Steklov., 243, 98–119 (2003).MathSciNetGoogle Scholar
  12. 12.
    C. K. Chui and P. W. Smith, “A note on Landau’s problem for bounded intervals,” Amer. Math. Monthly, 82, 927–929 (1975).MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    B. O. Eriksson, “Some best constants in the Landau inequality on a finite interval,” J. Approxim. Theory, 94, 420–452 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    A. M. Fink, “Kolmogorov–Landau inequalities for monotone functions,” J. Math. Appl., 90, 251–258 (1982).MathSciNetzbMATHGoogle Scholar
  15. 15.
    H. Kallioniemi, “The Landau problem on compact intervals and optimal numerical differentiation,” J. Approxim. Theory, 63, 72–91 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  17. 17.
    M. A. Krasnosel’skii and Ya. B. Rutitskii, Convex Functions and Orlicz Spaces [in Russian], Fizmatgiz, Moscow (1958).Google Scholar
  18. 18.
    A. Kroó and J. Szabados, “On the exact Markov inequality for k-monotone polynomials in uniform and L1-norm,” Acta Math. Hung., 125, No. 1-2, 99–112 (2009).zbMATHCrossRefGoogle Scholar
  19. 19.
    E. Landau, “Einige Ungleichungen fRur zweimal differenzierbane Funktion,” Proc. London Math. Soc., 13, 43–49 (1913).zbMATHCrossRefGoogle Scholar
  20. 20.
    G. V. Milovanović, D. S. Mitrinović, and T. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore (1994).zbMATHGoogle Scholar
  21. 21.
    N. Naidenov, “Landau-type extremal problem for the triple \( {\left\| f \right\|_\infty },\;{\left\| {f'} \right\|_p},\;{\left\| {f''} \right\|_\infty } \) on a finite interval,” J. Approxim. Theory, 123, 147–161 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    V. M. Olovyanishnikov, “On the question of the best inequalities between upper bounds of consecutive derivatives on a half-line,” Usp. Mat. Nauk, 6, 167–170 (1951).Google Scholar
  23. 23.
    M. K. Kwong, A. Zettl, Norm Inequalities for Derivatives and Differences, Springer, Berlin (1992).zbMATHGoogle Scholar
  24. 24.
    M. Sato, “The Landau inequality for bounded intervals with \( \left\| {{f^{(3)}}} \right\| \) finite,” J. Approxim. Theory, 34, 159–166 (1982).zbMATHCrossRefGoogle Scholar
  25. 25.
    A. Yu. Shadrin, “To the Landau–Kolmogorov problem on a finite interval,” in: Open Problems in Approximation Theory, SCT, Singapore (1994), pp. 192–204.Google Scholar
  26. 26.
    D. S. Skorokhodov, “On the Landau–Kolmogorov problem on an interval for absolutely monotone functions,” Vestn. Dnepropetr. Univ., Ser. Mat., 14, 120–128 (2009).Google Scholar
  27. 27.
    S. B. Stechkin, “Inequalities between norms of intermediate function derivatives,” Acta Sci. Math., 26, 225–230 (1965).zbMATHGoogle Scholar
  28. 28.
    S. B. Stechkin, “Best approximation of linear operators,” Math. Notes, 1, 91–99 (1967).MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Yu. N. Subbotin and N. I. Chernih, “Inequalities for derivatives of monotone functions,” in: Approximation of Functions. Theoretical and Applied Aspects (2003), pp. 199–211.Google Scholar
  30. 30.
    A. I. Zvyagintsev, “Strict inequalities for the derivatives of functions satisfying certain boundary conditions,” Math. Notes, 62, No. 5, 712–724 (1997).MathSciNetCrossRefGoogle Scholar
  31. 31.
    A. I. Zviagintsev and A. J. Lepin, “On the Kolmogorov inequalities between the upper bounds of function derivatives for n = 3,” Latv. Mat. Ezhegodnik, 26, 176–181 (1982).Google Scholar

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© Springer Science+Business Media New York 2012

Authors and Affiliations

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

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