### Kolmogorov inequalities for norms of Marchaud-type fractional derivatives of multivariate functions

#### Abstract

#### Keywords

#### Full Text:

PDF#### References

Arestov V.V. "Inequalities for fractional derivatives on the half-line", *Approximation theory: proc. conf.*, Warsaw, 1979; pp. 19-34.

Arestov V.V. "Approximation of unbounded operators by the bounded ones and related extremum problems", *Uspekhi mat. nauk*, 1996; 51(6): pp. 88-124. (in Russian) doi:10.1070/RM1996v051n06ABEH003001

Arestov V.V., Gabushin V.N. "The best approximation of unbounded operators by the bounded ones", *Izv. vuzov. Matematika*, 1995; 11: pp. 42-63. (in Russian)

Babenko V.F. "On sharp inequalities of Kolmogorov type for bivariate functions", *Dokl. NAN Ukrajiny*, 2000; 5: pp. 7-11. (in Russian)

Babenko V.F., Churilova M.S. "On inequalities of Kolmogorov type for derivatives of fractional order", *Res. Math.*, 2001; 6: pp. 16-20. (in Russian)

Babenko V.F., Kofanov V.A., Pichugov S.A. "Multivariate inequalities of Kolmogorov type and their applications", *Multivariate approximation and splines*, Basel, 1997; pp. 1-12. doi:10.1007/978-3-0348-8871-4_1

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

Babenko V.F., Matveeva T.V. "Kolmogorov-type inequalities for partial derivatives of multivariate functions", *Res. Math.*, 2008; 16: pp. 3-9 (in Russian)

Babenko V.F., Parfinovich N.V., Pichugov S.A. "Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions", *Ukrainian Math. J.*, 2010; 62(3): pp. 301-314. doi:10.1007/s11253-010-0358-y

Babenko V.F., Pichugov S.A. "Kolmogorov type inequalities for fractional derivatives of Holder functions of two variables", *East J. Approx*, 2007; 13(3): pp. 321-329.

Babenko V.F., Pichugov S.A. "Exact bounds for norms of fractional derivatives of the functions of several variables that satisfy Holder condition", *Mat. zametki*, 2010; 87(1): pp. 26-34. (in Russian) doi:10.1134/S0001434610010049

Buslaev A.P., Tikhomirov V.M. "On inequalities for derivatives in multidimensional case", *Mat. zametki*, 1979; 25(1): pp. 59-74. (in Russian) doi:10.1007/BF01138146

Dzyadyk V.K., Dubovik V.A. "To the Kolmogorov problem on dependence between upper bound of derivatives of real valued functions, defined on the whole real line", *Ukrainian Math. J.*, 1974; 26(3): pp. 246-259. (in Russian) doi:10.1007/BF01086215

Dzyadyk V.K., Dubovik V.A. "To Kolmogorov inequalities between upper bound of derivatives of real valued functions, defined on the whole real line", *Ukrainian Math. J.*, 1975; 27(3): pp. 291-299. (in Russian)

Gejsberg S.P. "Generalization of Hadamard inequality", *Researches on certain problems of constructive function theory: sb. nauch. tr. LOMI*, Leningrad, 1965; 50: pp. 42-54. (in Russian)

Kolmogorov A.N. "Une generalization de l’inegalite de M.J. Hadamard entre les bornes superieures des derivees successives d’une function", *C. r. Acad. sci. Paris*, 1938; 207: pp. 764-765. (in French)

Kolmogorov A.N. "On inequalities between upper bounds of consecutive derivatives of the function on infinite interval", *Uch. zap. MGU. Matematika*, 1939; 30(3): pp. 3-16. (in Russian)

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

Konovalov V.N. "Sharp inequalities for norms of functions, third partial and second mixed derivatives", *Matem. zametki*, 1978; 23(1): pp. 67-78. (in Russian) doi:10.1007/BF01104884

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

Magaril-Il’jaev G.G., Tihomirov V.M. "On the Kolmogorov inequality for fractional derivatives on the half-line", *Anal. Math.*, 1981; 7(1): pp. 37-47.

Mitrinović D.S., Pecârić J.E., Fink A.M. *Inequalities involving functions and their integrals and derivatives*, Kluver Acad. Publ., Dordrecht, Boston, London, 1991; 587 p.

Samko S.G., Kilbas A.A., Marichev O.I. *Integrals and derivatives of fractional order and their applications*, Nauka i tekhnika, Minsk, 1987; 688 p. (in Russian)

Stechkin S.B. "The best approximation of linear operators", *Matem. zametki*, 1967; 1(2): pp. 137-148. (in Russian) doi:10.1007/BF01268056

Timoshin O.A. "Sharp inequalities between the derivatives of the second and third order", *Dokl. RAN*, 1995; 344(1): pp. 20-22. (in Russian)

Rodov A.M. "Relations between upper bounds of derivatives of functions of a real variable", *Izv. AN SSSR. Ser. matem.*, 1946; 10: pp. 257-270. (in Russian)

Rodov A.M. "Sufficient conditions of existence of real values function with given upper bounds of moduli of the function itself and its five successive derivatives", *Uchenye Zap. Bel. Gos. Univ.*, 1954; 19: pp. 65-72. (in Russian)

DOI: https://doi.org/10.15421/242007

### Refbacks

- There are currently no refbacks.

Copyright (c) 2020 N.V. Parfinovych, V.V. Pylypenko

This work is licensed under a Creative Commons Attribution 4.0 International License.