Some integral inequalities in interval fractional calculus for left and right coordinated interval-valued functions

Muhammad Bilal Khan; Hatim Ghazi Zaini; Jorge E. Macías-Díaz; Savin Treanțǎ; Mohamed S. Soliman; Muhammad Bilal Khan; Hatim Ghazi Zaini; Jorge E. Macías-Díaz; Savin Treanțǎ; Mohamed S. Soliman

doi:10.3934/math.2022583

AIMS Mathematics

2022, Volume 7, Issue 6: 10454-10482. doi: 10.3934/math.2022583

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Some integral inequalities in interval fractional calculus for left and right coordinated interval-valued functions

1.
Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan
2.
Department of Computer Science, College of Computers and Information Technology, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
3.
Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico
4.
Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia
5.
Department of Applied Mathematics, University Politehnica of Bucharest, 060042 Bucharest, Romania
6.
Department of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 23 January 2022 Revised: 23 February 2022 Accepted: 11 March 2022 Published: 28 March 2022
MSC : Primary 26A33, 26A51, 26D07, 26D10; Secondary 26D15, 26D20

Integral inequalities play a crucial role in both theoretical and applied mathematics. Because of the relevance of these notions, we have discussed a new class of introduced generalized convex function called as coordinated left and right convex interval-valued function (coordinated LR-convex IVF) using the pseudo-order relation ($ {\le }_{p} $). On interval space, this order relation is defined. First, a pseudo-order relation is used to show Hermite-Hadamard type inequality (HH type inequality) for coordinated LR-convex IVF. Second for coordinated LR-convex IVF, Some HH type inequalities are also derived for the product of two coordinated LR-convex IVFs. Furthermore, we have demonstrated that our conclusions cover a broad range of new and well-known inequalities for coordinated LR-convex IVFs and their variant forms as special instances which are defined by Zhao et al. and Budak et al. Finally, we have shown that the inclusion relation "$ \supseteq $" confidents to the pseudo-order relation "$ {\le }_{p} $" for coordinated LR-convex IVFs. The concepts and methodologies presented in this study might serve as a springboard for additional research in this field, as well as a tool for investigating probability and optimization research, among other things.
Citation: Muhammad Bilal Khan, Hatim Ghazi Zaini, Jorge E. Macías-Díaz, Savin Treanțǎ, Mohamed S. Soliman. Some integral inequalities in interval fractional calculus for left and right coordinated interval-valued functions[J]. AIMS Mathematics, 2022, 7(6): 10454-10482. doi: 10.3934/math.2022583

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Abstract

Integral inequalities play a crucial role in both theoretical and applied mathematics. Because of the relevance of these notions, we have discussed a new class of introduced generalized convex function called as coordinated left and right convex interval-valued function (coordinated LR-convex IVF) using the pseudo-order relation ($ {\le }_{p} $). On interval space, this order relation is defined. First, a pseudo-order relation is used to show Hermite-Hadamard type inequality (HH type inequality) for coordinated LR-convex IVF. Second for coordinated LR-convex IVF, Some HH type inequalities are also derived for the product of two coordinated LR-convex IVFs. Furthermore, we have demonstrated that our conclusions cover a broad range of new and well-known inequalities for coordinated LR-convex IVFs and their variant forms as special instances which are defined by Zhao et al. and Budak et al. Finally, we have shown that the inclusion relation "$ \supseteq $" confidents to the pseudo-order relation "$ {\le }_{p} $" for coordinated LR-convex IVFs. The concepts and methodologies presented in this study might serve as a springboard for additional research in this field, as well as a tool for investigating probability and optimization research, among other things.

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