On inequalities of Kolmogorov type for classes of functions that are multiply monotonic on a finite interval

D.S. Skorokhodov

doi:10.15421/241018

On inequalities of Kolmogorov type for classes of functions that are multiply monotonic on a finite interval

D.S. Skorokhodov (Oles Honchar Dnipropetrovsk National University), https://orcid.org/0000-0001-8494-5885

Abstract

We solve the problem about exact constants in additive Kolmogorov-type inequalities on the class of multiply monotonic functions, defined on a finite interval.

Keywords

Kolmogorov type inequalities; multiply monotone functions

Full Text:

PDF (Русский)

References

Babenko Yu.V. "Pointwise inequalities of Landau-Kolmogorov type for functions defined on a finite segment", Ukrainian Math. J., 2001; 52(2): pp. 270-275. doi:10.1023/A:1010469004645

Bojanov B., Naidenov N. "Examples of Landau-Kolmogorov inequality in integral norms on a finite interval", J. Approx. Theory, 2002; 117(1): pp. 55-73. doi:10.1006/jath.2002.3687

Chui C.K., Smith P.W. "A note on Landau's problem for bounded intervals", Amer. Math. Monthly, 1975; 82(9): pp. 927-929. doi:10.1080/00029890.1975.11993983

Fink A.M. "Kolmogorov-Landau inequalities for monotone functions", J. Math. Appl., 1992; 90(1): pp. 251-258. doi:10.1016/0022-247X(82)90057-9

Landau E. "Einige Ungleichungen für zweimal differentiierbare Funktionen", Proc. London Math. Soc., 1913; 13(2): pp. 43-49. (in German) doi:10.1112/plms/s2-13.1.43

Naidenov N. "Landau-type extremal problem for the triple $$$\| f \|_{\infty}$$$, $$$\| f' \|_p$$$, $$$\| f'' \|_{\infty}$$$ on a finite interval", J. Approx. Theory, 2003; 123(2): pp. 147-161. doi:10.1016/S0021-9045(03)00081-9

Sato M. "The Landau inequality for bounded intervals with $$$f^{(3)}$$$ finite", J. Approx. Theory, 1982; 34(2): pp. 159-166. doi:10.1016/0021-9045(82)90089-2

Shadrin A.Yu. "To the Landau-Kolmogorov problem on a finite interval", Proc. Int. Conf. "Open Problems in Approximation Theory", Vonesta Voda, 1993; pp. 192-204.

Babenko V.F., Korneichuk N.P., Kofanov V.A., Pichugov S.A. Inequalities for derivatives and their applications, Naukova dumka, 2003.

Zvyagintsev A.I., Lepin A.Ya. "On exact Kolmogorov inequalities between upper bounds of function derivatives in the case $n = 3$", Latvian Matem. Ezhegodnik, 1982; 26: pp. 176-181.

Krasnoselskii M.A., Rutitskii Ya.B. Convex functions and Orlich spaces, 1958;

Krein S.G., Petunin Yu.I., Semyonov Ye.M. Interpolation of linear operators, Nauka, 1978.

Skorokhodov D.S. "On Landau-Kolmogorov problem on the segment for absolutely monotonic functions", Res. Math., 2009; 14: pp. 120-128.