Ukrainian Mathematical Journal

, Volume 64, Issue 5, pp 721–736 | Cite as

Inequalities for derivatives of functions on an axis with nonsymmetrically bounded higher derivatives

  • V. A. Kofanov

For nonperiodic functions \( x\in L_{\infty}^r(R) \) defined on the entire real axis, we prove analogs of the Babenko inequality. The obtained inequalities estimate the norms of derivatives \( \left\| {x_{\pm}^{(k) }} \right\|{L_{q[a,b] }} \) on an arbitrary interval [a, b] ⊂ R such that x (k)(a) = x (k)(b) = 0 via local L p -norms of the functions x and uniform nonsymmetric norms of the higher derivatives x(r) of these functions.


Arbitrary Function Comparison Theorem High Derivative Yield Inequality Account Relation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. A. Ligun, “Inequalities for upper bounds of functionals,” Anal. Math., 2, No. 1, 11–40 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    V. F. Babenko, “Nonsymmetric extremal problems in approximation theory,” Dokl. Akad. Nauk SSSR, 269, No. 3, 521–524 (1983).MathSciNetGoogle Scholar
  3. 3.
    B. Bojanov and N. Naidenov, “An extension of the Landau–Kolmogorov inequality. Solution of a problem of Erdos,” J. d’Anal. Math., 78, 263–280 (1999).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    V. A. Kofanov, “Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function,” Ukr. Mat. Zh., 63, No. 7, 969–984 (2011); English translation: Ukr. Math. J., 63, No. 7, 1118–1135 (2011).CrossRefGoogle Scholar
  5. 5.
    A. Pinkus and O. Shisha, “Variations on the Chebyshev and Lq theories of best approximation,” J. Approxim. Theory, 35, No. 2, 148–168 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    L. Hörmander, “A new proof and generalization of inequality of Bohr,” Math. Scand., 2, 33–45 (1954).MathSciNetzbMATHGoogle Scholar
  7. 7.
    N. P. Korneichuk, V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, Inequalities for Derivatives and Their Applications [in Russian], Naukova Dumka, Kiev (2003).Google Scholar
  8. 8.
    G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University, Cambridge (1934).Google Scholar
  9. 9.
    N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  10. 10.
    V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Inequalities of Kolmogorov type and some their applications in approximation theory,” Rend. Circ. Mat. Palermo, Ser. II, Suppl., 52, 223–237 (1998).Google Scholar
  11. 11.
    V. A. Kofanov, “Some exact inequalities of Kolmogorov type,” Mat. Fiz., Anal., Geom., 9, No. 3, 1–8 (2002).Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • V. A. Kofanov
    • 1
  1. 1.Dnepropetrovsk National UniversityDnepropetrovskUkraine

Personalised recommendations