Vietnam Journal of Mathematics

, Volume 43, Issue 1, pp 173–179 | Cite as

Some Extensions of the Kolmogorov–Stein Inequality

  • Ha Huy BangEmail author
  • Vu Nhat Huy


In this paper, we prove some extensions of the Kolmogorov–Stein inequality for derivatives in L p (ℝ) norm to differential operators generated by a polynomial.


Lp spaces Orlicz spaces Kolmogorov inequality 

Mathematics Subject Classification (2010)

26A24 41A17 



This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.32.


  1. 1.
    Bang, H.H.: A remark on the Kolmogorov–Stein inequality. J. Math. Anal. Appl. 203, 861–867 (1996)Google Scholar
  2. 2.
    Bang, H.H.: On an inequality of Bohr for Orlicz spaces. Bull. Pol. Acad. Sci. 49, 383–389 (2001)Google Scholar
  3. 3.
    Bang, H.H., Thu, M.T.: On a Landau–Kolmogorov inequality. J. Inequal. Appl. 7, 663–672 (2002)Google Scholar
  4. 4.
    Bohr, H.: Ein allgemeiner Satz über die Integration eines trigonometrischen Polynoms. Prace Matem. -Fiz. 43, 273–288 (1935)Google Scholar
  5. 5.
    Bojanov, B.D., Varma, A.K.: On a polynomial inequality of Kolmogorov type. Proc. Amer. Math. Soc. 124, 491–496 (1996)Google Scholar
  6. 6.
    Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Graduate Texts in Mathematics, vol. 161. Springer-Verlag, New York (1995)Google Scholar
  7. 7.
    Burenkov, V.I.: Exact constants in inequalities for norms of intermediate derivatives on a finite interval. Proc. Steklov Inst. Math. 156, 22–29 (1980)Google Scholar
  8. 8.
    Certain, M.W., Kurtz, T.G.: Landau–Kolmogorov inequalities for semigroups and groups. Proc. Amer. Math. Soc. 63, 226–230 (1977)Google Scholar
  9. 9.
    Chernov, P.R.: Optimal Landau–Kolmogorov inequalities for dissipative operators in Hilbert and Banach spaces. Adv. Math. 34, 137–144 (1979)Google Scholar
  10. 10.
    Ditzian, Z.: Some remarks on inequalities of Landau and Kolmogorov. Aequat. Math. 12, 145–151 (1975)Google Scholar
  11. 11.
    Ditzian, Z.: A Kolmogorov-type inequality. Math. Proc. Camb. Philos. Soc. 136, 657–663 (2004)Google Scholar
  12. 12.
    Favard, J.: Application de la formule sommatoire d’Euler ` la démonstration de quelques propriétés extrémales des intégrales des fonctions périodiques et presque-périodiques. Mat. Tidsskr. B, 81–94 (1936)Google Scholar
  13. 13.
    Hörmander, L.: A new generalization of an inequality of Bohr. Math. Scand. 2, 33–45 (1954)Google Scholar
  14. 14.
    Kofanov, V.A.: On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives. Ukr. Math. J. 55, 548–565 (2008)Google Scholar
  15. 15.
    Kolmogorov, A.N.: On inequalities between upper bounds of the successive derivatives of an arbitrary function on an infinite interval. Amer. Math. Soc. Transl. Ser. 1 2, 233–243 (1962)Google Scholar
  16. 16.
    Krasnoselskii, M.A., Rutickii, Y.B.: Convex functions and orlicz spaces. GITTL, Moscow (1958). Engl. Transl. Noordhoff (1961)Google Scholar
  17. 17.
    Luxemburg, W.: Banach function spaces. (Thesis). Technische Hogeschool te Delft., The Netherlands (1955)Google Scholar
  18. 18.
    Nikolskii, S.M.: Approximation of functions of several variables and imbedding theorems. Moscow, Nauka (1977)Google Scholar
  19. 19.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces. Marcel Dekker, New York (1991)zbMATHGoogle Scholar
  20. 20.
    Stein, E.M.: Functions of exponential type. Ann. Math. 65, 582–592 (1957)Google Scholar
  21. 21.
    Tikhomirov, V.M., Magaril-Il’jaev, G.G.: Inequalities for derivatives. In Kolmogorov, A.N. Selected Papers, pp 387–390. Moscow, Nauka (1985)Google Scholar
  22. 22.
    Trigub, R.M.: Comparison of linear differential operators. Math. Notes 82, 380–394 (2007)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014

Authors and Affiliations

  1. 1.Institute of MathematicsVietnamese Academy of Science and TechnologyHanoiVietnam
  2. 2.Department of Mathematics, College of ScienceVietnam National UniversityHanoiVietnam

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