Inequalities of Simpson-Mercer-type including Atangana-Baleanu fractional operators and their applications

Muhammad Tariq; Hijaz Ahmad; Soubhagya Kumar Sahoo; Artion Kashuri; Taher A. Nofal; Ching-Hsien Hsu; Muhammad Tariq; Hijaz Ahmad; Soubhagya Kumar Sahoo; Artion Kashuri; Taher A. Nofal; Ching-Hsien Hsu

doi:10.3934/math.2022831

AIMS Mathematics

2022, Volume 7, Issue 8: 15159-15181. doi: 10.3934/math.2022831

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Research article

Inequalities of Simpson-Mercer-type including Atangana-Baleanu fractional operators and their applications

1.
Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan
2.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, Roma 00186, Italy
3.
Department of Mathematics, Department of Mathematics, Institute of Technical Education and Research, Siksha 'O' Anusandhan University, Bhubaneswar 751030, India
4.
Department of Mathematics, Faculty of Technical and Natural Sciences, University Ismail Qemali, Vlora 9400, Albania
5.
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
6.
Department of Computer Science and Information Engineering, Asia University, Taiwan
7.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
8.
Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology, School of Mathematics and Big Data, Foshan University, Foshan 528000, China

Received: 27 December 2021 Revised: 06 May 2022 Accepted: 11 May 2022 Published: 16 June 2022
MSC : 26A51, 26A33, 26D10

Fractional operators with integral inequalities have attracted the interest of several mathematicians. Fractional inequalities are best utilized in mathematical science with their features and wide range of applications in optimization, modeling, engineering and artificial intelligence. In this article, we consider new variants of Simpson-Mercer type inequalities involving the Atangana-Baleanu (A-B) fractional integral operator for $ s $-convex functions. First, an integral identity, which acts as an auxiliary result for the main results is proved in the frame of fractional operator. Employing this new identity, some estimations of Simpson-Mercer type for $ s $-convex functions in the second sense are discussed. In addition, we study various new applications on Modified Bessel functions, special means and $ q $-digamma functions. These applications confirm the effectiveness and validity of the results and also bring a different dimension to the study.
- Jensen-Mercer inequality,
- Simpson's inequality,
- $ s $-convex functions,
- Atangana-Baleanu fractional operator,
- Hölder's inequalities,
- q-digamma functions,
- Modified Bessel functions
Citation: Muhammad Tariq, Hijaz Ahmad, Soubhagya Kumar Sahoo, Artion Kashuri, Taher A. Nofal, Ching-Hsien Hsu. Inequalities of Simpson-Mercer-type including Atangana-Baleanu fractional operators and their applications[J]. AIMS Mathematics, 2022, 7(8): 15159-15181. doi: 10.3934/math.2022831

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Abstract

Fractional operators with integral inequalities have attracted the interest of several mathematicians. Fractional inequalities are best utilized in mathematical science with their features and wide range of applications in optimization, modeling, engineering and artificial intelligence. In this article, we consider new variants of Simpson-Mercer type inequalities involving the Atangana-Baleanu (A-B) fractional integral operator for $ s $-convex functions. First, an integral identity, which acts as an auxiliary result for the main results is proved in the frame of fractional operator. Employing this new identity, some estimations of Simpson-Mercer type for $ s $-convex functions in the second sense are discussed. In addition, we study various new applications on Modified Bessel functions, special means and $ q $-digamma functions. These applications confirm the effectiveness and validity of the results and also bring a different dimension to the study.

References

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