On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities

Miguel Vivas-Cortez; Pshtiwan O. Mohammed; Y. S. Hamed; Artion Kashuri; Jorge E. Hernández; Jorge E. Macías-Díaz; Miguel Vivas-Cortez; Pshtiwan O. Mohammed; Y. S. Hamed; Artion Kashuri; Jorge E. Hernández; Jorge E. Macías-Díaz

doi:10.3934/math.2022571

AIMS Mathematics

2022, Volume 7, Issue 6: 10256-10275. doi: 10.3934/math.2022571

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On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities

1.
Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Católica del Ecuador, Ecuador
2.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq
3.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
4.
Department of Mathematics, Faculty of Technical Science, University "Ismail Qemali", Vlora 9400, Albania
5.
Departamento de Técnicas Cuantitativas, Universidad Centroccidental Lisandro Alvarado, Venezuela
6.
Department of Mathematics and Didactics of Mathematics, Tallinn University, Estonia
7.
Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Mexico

Received: 08 December 2021 Revised: 07 March 2022 Accepted: 13 March 2022 Published: 23 March 2022
MSC : 26D15, 26D10, 26A33

In this work, we introduce generalized Raina fractional integral operators and derive Chebyshev-type inequalities involving these operators. In a first stage, we obtain Chebyshev-type inequalities for one product of functions. Then we extend those results to account for arbitrary products. Also, we establish some inequalities of the Chebyshev type for functions whose derivatives are bounded. In addition, we derive an estimate for the Chebyshev functional by applying the generalized Raina fractional integral operators. As corollaries of this study, some known results are recaptured from our general Chebyshev inequalities. The results of this work may prove useful in the theoretical analysis of numerical models to solve generalized Raina-type fractional-order integro-differential equations.
- Chebyshev inequality,
- generalized Raina integral operators,
- integral inequalities,
- fractional-order integrals,
- approximation techniques
Citation: Miguel Vivas-Cortez, Pshtiwan O. Mohammed, Y. S. Hamed, Artion Kashuri, Jorge E. Hernández, Jorge E. Macías-Díaz. On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities[J]. AIMS Mathematics, 2022, 7(6): 10256-10275. doi: 10.3934/math.2022571

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Abstract

In this work, we introduce generalized Raina fractional integral operators and derive Chebyshev-type inequalities involving these operators. In a first stage, we obtain Chebyshev-type inequalities for one product of functions. Then we extend those results to account for arbitrary products. Also, we establish some inequalities of the Chebyshev type for functions whose derivatives are bounded. In addition, we derive an estimate for the Chebyshev functional by applying the generalized Raina fractional integral operators. As corollaries of this study, some known results are recaptured from our general Chebyshev inequalities. The results of this work may prove useful in the theoretical analysis of numerical models to solve generalized Raina-type fractional-order integro-differential equations.

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