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Bojanov–Naidenov Problem for Functions with Asymmetric Restrictions for the Higher Derivative

  • V. A. KofanovEmail author
Article
For given r ∈ N, p, 𝛼, β,μ > 0, we solve the extreme problems
$$ \underset{a}{\overset{b}{\int }}{x}_{\pm}^q(t) dt\to \sup, \kern0.5em q\ge p, $$

in the set of pairs (x, I) of functions \( x\in {L}_{\infty}^r \) and intervals I = [a, b] R satisfying the inequalities −βx(r)(t) ≤ 𝛼 for almost all tR, the conditions\( L{\left({x}_{\pm}\right)}_p\le L{\left({\left({\varphi}_{\leftthreetimes, r}^{\alpha, \beta}\right)}_{\pm}\right)}_p \), and the corresponding condition μ(supp[a, b]x+) ≤ μ or μ(supp[a, b]x−) ≤ μ, where L(x)p; = sup {‖xLp[a, b]; a, b ∈ R,  |x(t)| > 0, t ∈ (a, b)}, supp[a, b]x ±  ≔ {t ∈ [a, b] : x±(t) > 0},, and \( {\varphi}_{\leftthreetimes, r}^{\alpha, \beta } \) is an asymmetric (2π/⋋) -periodic Euler spline of order r. As a consequence, we solve the same extreme problems for the intermediate derivatives \( {x}_{\pm}^{(k)} \), k = 1, . . . , r − 1, with q ≥ 1.

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References

  1. 1.
    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
  2. 2.
    V. F. Babenko, “Investigations of Dnepropetrovsk mathematicians related to inequalities for the derivatives of periodic functions and their applications,” Ukr. Mat. Zh., 52, No. 1, 5–29 (2000); English translation: Ukr. Math. J., 52, No. 1, 8–28 (2000).CrossRefGoogle Scholar
  3. 3.
    M. K. Kwong and A. Zettl, Norm Inequalities for Derivatives and Differences, Springer, Berlin (1992).CrossRefGoogle Scholar
  4. 4.
    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).MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Pinkus and O. Shisha, “Variations on the Chebyshev and L q theories of best approximation,” J. Approxim. Theory, 35, No. 2, 148–168 (1982).CrossRefGoogle Scholar
  6. 6.
    V. A. Kofanov, “On some extremal problems of different metrics for differentiable functions on the axis,” Ukr. Mat. Zh., 61, No. 6, 765–776 (2009); English translation: Ukr. Math. J., 61, No. 6, 908–922 (2009).MathSciNetCrossRefGoogle Scholar
  7. 7.
    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).MathSciNetCrossRefGoogle Scholar
  8. 8.
    L. H¨ormander, “A new proof and generalization of the inequality of Bohr,” Math. Scand., 2, 33–45 (1954).MathSciNetCrossRefGoogle Scholar
  9. 9.
    V. F. Babenko, “Nonsymmetric extremal problems of approximation theory,” Dokl. Akad. Nauk SSSR, 269, No. 3, 521–524 (1983).MathSciNetGoogle 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, “Inequalities for the derivatives of functions on an axis with nonsymmetrically bounded higher derivatives,” Ukr. Mat. Zh., 64, No 5, 636–648 (2012); English translation: Ukr. Math. J., 64, No 5, 721–736 (2012).Google Scholar
  12. 12.
    V. V. Kameneva and V. A. Kofanov, “The Bojanov–Naidenov problem for positive (negative) parts of functions differentiable on the axis,” Visn. Dnipr. Univ., Ser. Mat., Issue 23, 25–36 (2018).CrossRefGoogle Scholar
  13. 13.
    N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Dnepropetrovsk National UniversityDniproUkraine

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