An analysis of fractional integral calculus and inequalities by means of coordinated center-radius order relations

Waqar Afzal; Mujahid Abbas; Jongsuk Ro; Khalil Hadi Hakami; Hamad Zogan; Waqar Afzal; Mujahid Abbas; Jongsuk Ro; Khalil Hadi Hakami; Hamad Zogan

doi:10.3934/math.20241499

AIMS Mathematics

2024, Volume 9, Issue 11: 31087-31118. doi: 10.3934/math.20241499

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An analysis of fractional integral calculus and inequalities by means of coordinated center-radius order relations

1.
Department of Mathematics, Government College University, Katchery Road, Lahore 54000, Pakistan
2.
Department of Mechanical Engineering Sciences, Faculty of Engineering and the Built Environment, Doornfontein Campus, University of Johannesburg, South Africa
3.
Department of Medical Research, China Medical University, Taichung 406040, Taiwan
4.
School of Electrical and Electronics Engineering, Chung-Ang University, Dongjak-gu, Seoul 06974, Korea
5.
Department of Intelligent Energy and Industry, Chung-Ang University, Dongjak-gu, Seoul 06974, Korea
6.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Saudi Arabia
7.
Department of Computer Science, College of Engineering and Computer Science, Jazan University, Jazan, Saudi Arabia

Received: 21 August 2024 Revised: 12 October 2024 Accepted: 18 October 2024 Published: 31 October 2024
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

Interval-valued maps adjust integral inequalities using different types of ordering relations, including inclusion and center-radius, both of which behave differently. Our purpose was to develop various novel bounds and refinements for weighted Hermite-Hadamard inequalities as well as their product form by employing new types of fractional integral operators under a cr-order relation. Mostly authors have used inclusion order to adjust inequalities in interval maps, but they have some flaws, specifically they lack the property of comparability between intervals. However, we show that under cr-order, it satisfies all relational properties of intervals, including reflexivity, antisymmetry, transitivity, and comparability and preserves integrals as well. Furthermore, we provide numerous interesting remarks, corollaries, and examples in order to demonstrate the accuracy of our findings.
- weighted Hermite-Hadamard,
- symmetric mappings,
- fractional calculus,
- coordinated cr-order
Citation: Waqar Afzal, Mujahid Abbas, Jongsuk Ro, Khalil Hadi Hakami, Hamad Zogan. An analysis of fractional integral calculus and inequalities by means of coordinated center-radius order relations[J]. AIMS Mathematics, 2024, 9(11): 31087-31118. doi: 10.3934/math.20241499

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Abstract

Interval-valued maps adjust integral inequalities using different types of ordering relations, including inclusion and center-radius, both of which behave differently. Our purpose was to develop various novel bounds and refinements for weighted Hermite-Hadamard inequalities as well as their product form by employing new types of fractional integral operators under a cr-order relation. Mostly authors have used inclusion order to adjust inequalities in interval maps, but they have some flaws, specifically they lack the property of comparability between intervals. However, we show that under cr-order, it satisfies all relational properties of intervals, including reflexivity, antisymmetry, transitivity, and comparability and preserves integrals as well. Furthermore, we provide numerous interesting remarks, corollaries, and examples in order to demonstrate the accuracy of our findings.

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