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Landau–Kolmogorov problem for a class of functions absolutely monotone on a finite interval

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We solve the Landau–Kolmogorov problem for a class of functions absolutely monotone on a finite interval. For this class of functions, new exact additive inequalities of the Kolmogorov type are obtained.

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References

  1. E. Landau, “Einige Ungleichungen f¨ur zweimal differenzierbare Funktionen,” Proc. London Math. Soc., 13, 43–49 (1913).

    Article  MATH  Google Scholar 

  2. C. K. Chui and P. W. Smith, “A note on Landau’s problem for bounded intervals,” Amer. Math. Mon., 82, 927–929 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. I. Zvyagintsev and A. Ya. Lepin, “On Kolmogorov inequalities between upper bounds of functional derivatives for n = D 3,” Latv. Mat. Ezhegod., 26, 176–181 (1982).

    MathSciNet  MATH  Google Scholar 

  4. M. Sato, “The Landau inequality for bounded intervals with ∥f (3)∥ finite,” J. Approxim. Theory, 34, 159–166 (1982).

    Article  MATH  Google Scholar 

  5. M. K. Kwong and A. Zettl, Norm Inequalities for Derivatives and Differences, Springer, Berlin (1992).

    MATH  Google Scholar 

  6. A. Yu. Shadrin, “To the Landau–Kolmogorov problem on a finite interval,” in: B. Bojanov (editor), Open Problems in Approximation Theory, SCT, Singapore (1994), pp. 192–204.

    Google Scholar 

  7. B. Bojanov and N. Naidenov, “An extension of the Landau–Kolmogorov inequality. Solution of a problem of Erdős,” J. D’Anal. Math., 78, 263–280 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Bojanov and N. Naidenov, “Examples of Landau–Kolmogorov inequality in integral norms on a finite interval,” J. Approxim. Theory, 117, 55–73 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  9. Yu. V. Babenko, “Exact inequalities of Landau type for functions with second derivatives from the Orlicz space,” Vestn. Dnepropetr. Univ., Ser. Mat., 2, 18–22 (2000).

    Google Scholar 

  10. Yu. V. Babenko, “Pointwise inequalities of Landau–Kolmogorov type for functions defined on a finite segment,” Ukr. Mat. Zh., 53, No. 2, 238–243 (2001); English translation: Ukr. Math. J., 53, No. 2, 270–275 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  11. N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).

    Google Scholar 

  12. 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 

  13. V. F. Babenko and T. M. Rassias, “On exact inequalities of Hardy–Littlewood–Pólya type,” J. Math. Anal. Appl., 245, 570–593 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. M. Fink, “Kolmogorov–Landau inequalities for monotone functions,” J. Math. Appl., 90, 251–258 (1992).

    MathSciNet  Google Scholar 

  15. D. V. Widder, The Laplace Transform, Princeton University, Princeton (1946).

    Google Scholar 

  16. D. S. Skorokhodov, “On the Landau–Kolmogorov problem for classes of functions absolutely monotone on a segment,” in: Abstracts of the Mathematical Congress (August 27–29, 2009, Kyiv) [in Ukrainian], Kyiv (2009).

  17. M. G. Krein and A. A. Nudel’man, Markov Moment Problem and Extremal Problems [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  18. A. N. Kolmogorov, “On inequalities between upper bounds of successive derivatives of an arbitrary function on an infinite interval,” Nauch. Zap. Mosk. Univ., 30, 3–16 (1939).

    Google Scholar 

  19. A. N. Kolmogorov, “On inequalities between upper bounds of successive derivatives of an arbitrary function on an infinite interval,” in: A. N. Kolmogorov, Selected Works. Mathematics and Mechanics [in Russian] (1985), pp. 252–263.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 4, pp. 531–548, April, 2011.

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Skorokhodov, D.S. Landau–Kolmogorov problem for a class of functions absolutely monotone on a finite interval. Ukr Math J 63, 617–637 (2011). https://doi.org/10.1007/s11253-011-0529-5

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  • DOI: https://doi.org/10.1007/s11253-011-0529-5

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