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),


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 $$$\|x^{(r)}\|_{\infty} \leqslant 1$$$, $$$\|x\|_{p,\delta} \leqslant \| \varphi_{\lambda,r} \|_{p,\delta}$$$, $$$\delta \in (0,\pi / \lambda)$$$.


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


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)

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.

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.

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)

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

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. (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)




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