### Solution of Bojanov-Naidenov problem with constraints for the norm $\|x\|_{p,\delta} = \sup \bigl\{ \| x \|_{L_p[a;b]} \colon a,b\in \mathbb{R}, b-a\leqslant \delta \bigr\}$

V.A. Kofanov (Oles Honchar Dnipropetrovsk National University), https://orcid.org/0000-0003-0392-2257

#### Abstract

For given $r\in \mathbb{N}$; $p,\lambda > 0$ and fixed interval $[a;b] \subset \mathbb{R}$ we solve the extremal problems 1) $\int\limits_a^b |x(t)|^q dt \rightarrow \sup$, $q > p$, 2) $\int\limits_a^b |x^{(k)}(t)|^q dt \rightarrow \sup$, $q \geqslant 1$, $k\in \mathbb{N}$, $k < r$, on the set of functions $f\in L^r_{\infty}$ such that $\|x^{(r)}\|_{\infty} \leqslant 1$, $\|x\|_{p,\delta} \leqslant \| \varphi_{\lambda,r} \|_{p,\delta}$, $\delta \in (0,\pi / \lambda)$.

#### Keywords

inequalities of various metrics; Kolmogorov-type inequalities; Bojanov-Naidenov problem

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#### References

Korneichuk N.P., Babenko V.F., Kofanov V.A., Pichugov S.A. Inequalities for derivatives and their applications, Nauk. dumka, Kyiv, 2003; 590 p. (in Russian)

Babenko V.F. "Researches of Dnipropetrovsk mathematicians on inequalities for derivatives of periodic functions and their applications", Ukrainian Math. J., 2000, 52(1): pp. 5-29. (in Russian) doi:10.1007/BF02514133

Kwong M.K., Zettl A. "Norm inequalities for derivatives and differences", Lecture notes in mathematics, 1993; 1536: 150 p.

Bojanov B., Naidenov N. "An extension of the Landau-Kolmogorov inequality. Solution of a problem of Erdös", J. d'Analyse Mathematique, 1999; 78: pp. 263-280. doi:10.1007/BF02791137

Pinkus A., Shisha O. "Variations on the Chebyshev and $L^q$-Theories of Best Approximation", Journal of Approx. Theory, 1982; 35(2): pp. 148-168. doi:10.1016/0021-9045(82)90033-8

Kofanov V.A. "Some extremum problems in different metrics for differentiable functions on the real domain", Ukrainian Math. J., 2009; 61(6): pp. 765-776. (in Russian) doi:10.1007/s11253-009-0254-5

Kofanov V.A. "Inequalities of various metrics for differentiable periodic functions", Ukrainian Math. J., 2015; 67(2): pp. 202-212. (in Russian) doi:10.1007/s11253-015-1076-2

Kolmogorov A.N. "On inequalities between upper bounds of consecutive derivatives of the function on infinite interval", Izbr. tr. Matematika, mekhanika, Nauka, Moscow, 1985; pp. 252-263. (in Russian)

Korneichuk N.P., Babenko V.F., Ligun A.A. Extremum properties of polynomials and splines, Naukova dumka, Kyiv, 1992; 304 p. (in Russian)

Kofanov V.A. "Sharp inequalities of Bernstein and Kolmogorov type", East J. Approx., 2005; 11(2): pp. 131-145.

Korneichuk N.P., Ligun A.A., Doronin V.G. Approximation with constraints, Kyiv, 1982; 250 p. (in Ukrainian)

DOI: https://doi.org/10.15421/241705

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##### ISSN (Print): 2664-4991 