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Some Extensions of the Kolmogorov–Stein Inequality

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Abstract

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.

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References

  1. Bang, H.H.: A remark on the Kolmogorov–Stein inequality. J. Math. Anal. Appl. 203, 861–867 (1996)

  2. Bang, H.H.: On an inequality of Bohr for Orlicz spaces. Bull. Pol. Acad. Sci. 49, 383–389 (2001)

  3. Bang, H.H., Thu, M.T.: On a Landau–Kolmogorov inequality. J. Inequal. Appl. 7, 663–672 (2002)

  4. Bohr, H.: Ein allgemeiner Satz über die Integration eines trigonometrischen Polynoms. Prace Matem. -Fiz. 43, 273–288 (1935)

    Google Scholar 

  5. Bojanov, B.D., Varma, A.K.: On a polynomial inequality of Kolmogorov type. Proc. Amer. Math. Soc. 124, 491–496 (1996)

  6. Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Graduate Texts in Mathematics, vol. 161. Springer-Verlag, New York (1995)

  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)

  8. Certain, M.W., Kurtz, T.G.: Landau–Kolmogorov inequalities for semigroups and groups. Proc. Amer. Math. Soc. 63, 226–230 (1977)

  9. Chernov, P.R.: Optimal Landau–Kolmogorov inequalities for dissipative operators in Hilbert and Banach spaces. Adv. Math. 34, 137–144 (1979)

  10. Ditzian, Z.: Some remarks on inequalities of Landau and Kolmogorov. Aequat. Math. 12, 145–151 (1975)

  11. Ditzian, Z.: A Kolmogorov-type inequality. Math. Proc. Camb. Philos. Soc. 136, 657–663 (2004)

  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)

  13. Hörmander, L.: A new generalization of an inequality of Bohr. Math. Scand. 2, 33–45 (1954)

  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)

  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)

  16. Krasnoselskii, M.A., Rutickii, Y.B.: Convex functions and orlicz spaces. GITTL, Moscow (1958). Engl. Transl. Noordhoff (1961)

  17. Luxemburg, W.: Banach function spaces. (Thesis). Technische Hogeschool te Delft., The Netherlands (1955)

    Google Scholar 

  18. Nikolskii, S.M.: Approximation of functions of several variables and imbedding theorems. Moscow, Nauka (1977)

    Google Scholar 

  19. Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces. Marcel Dekker, New York (1991)

    MATH  Google Scholar 

  20. Stein, E.M.: Functions of exponential type. Ann. Math. 65, 582–592 (1957)

  21. Tikhomirov, V.M., Magaril-Il’jaev, G.G.: Inequalities for derivatives. In Kolmogorov, A.N. Selected Papers, pp 387–390. Moscow, Nauka (1985)

  22. Trigub, R.M.: Comparison of linear differential operators. Math. Notes 82, 380–394 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

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

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Authors and Affiliations

  1. Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet Street, Cau Giay, Hanoi, Vietnam

    Ha Huy Bang

  2. Department of Mathematics, College of Science, Vietnam National University, 334 Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam

    Vu Nhat Huy

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  1. Ha Huy Bang
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  2. Vu Nhat Huy
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Correspondence to Ha Huy Bang.

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Bang, H.H., Huy, V.N. Some Extensions of the Kolmogorov–Stein Inequality. Vietnam J. Math. 43, 173–179 (2015). https://doi.org/10.1007/s10013-014-0090-2

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  • DOI: https://doi.org/10.1007/s10013-014-0090-2

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