This paper investigated new Hermite-Hadamard-Mercer type inequalities associated with an Atangana-Baleanu-conformable fractional integral operator. By combining the structural features of Atangana-Baleanu fractional integrals and conformable kernels, we derived a fractional framework that contained both local and nonlocal effects. A fundamental identity was first established, transforming a symmetric combination of endpoint values and fractional integral terms into a weighted integral involving the first derivative. Based on this identity and Jensen-Mercer's inequality, several new bounds were obtained under convexity assumptions on $ |f^{\prime }| $ and $ |f^{\prime}|^{q} $, where $ q > 1 $. The results extended known Hermite-Hadamard-Mercer inequalities and reduced to classical or fractional special cases under suitable parameter choices. The proposed approach provided a flexible tool for fractional integral inequalities and related estimates in convex analysis.
Citation: Jen Chieh Lo. Some Hermite-Hadamard-Mercer type inequalities via Atangana-Baleanu-conformable fractional operator[J]. AIMS Mathematics, 2026, 11(6): 17937-17950. doi: 10.3934/math.2026731
This paper investigated new Hermite-Hadamard-Mercer type inequalities associated with an Atangana-Baleanu-conformable fractional integral operator. By combining the structural features of Atangana-Baleanu fractional integrals and conformable kernels, we derived a fractional framework that contained both local and nonlocal effects. A fundamental identity was first established, transforming a symmetric combination of endpoint values and fractional integral terms into a weighted integral involving the first derivative. Based on this identity and Jensen-Mercer's inequality, several new bounds were obtained under convexity assumptions on $ |f^{\prime }| $ and $ |f^{\prime}|^{q} $, where $ q > 1 $. The results extended known Hermite-Hadamard-Mercer inequalities and reduced to classical or fractional special cases under suitable parameter choices. The proposed approach provided a flexible tool for fractional integral inequalities and related estimates in convex analysis.
| [1] |
E. Set, A. O. Akdemir, A. Karaoĝlan, T. Abdeljawad, W. Shatanawi, On new generalizations of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional integral operator, Axioms, 10 (2021), 223. https://doi.org/10.3390/axioms10030223 doi: 10.3390/axioms10030223
|
| [2] |
E. Set, S. I. Butt, A. O. Akdemir, A. Karaoglan, T. Abdeljaward, New integral inequalities for differentiable convex functions via Atangana-Baleanu fractional integral operators, Chaos Solitons Fract., 143 (2021), 110554. https://doi.org/10.1016/j.chaos.2020.110554 doi: 10.1016/j.chaos.2020.110554
|
| [3] |
J. B. Jiu, S. I. Butt, J. Nasir, A. Aslam, A. Fahad, J. Soontharanon, Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator, AIMS Math., 7 (2022), 2123–2141. https://doi.org/10.3934/math.2022121 doi: 10.3934/math.2022121
|
| [4] | M. Tariq, S. K. Ntouyas, H. Ahmad, R. Efendiev, Some new modifications of integral inequalities in the frame of AB fractional operator, Trans. Natl. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci. Mathematics, 46 (2026), 1–22. |
| [5] |
M. Karim, A. Fahmi, Z. Ullah, M. A. T. Bhatti, A. Qayyum, On certain Ostrowski type integral inequalities for convex function via AB-fractional operator, AIMS Math., 8 (2023), 9166–9184. https://doi.org/10.3934/math.2023459 doi: 10.3934/math.2023459
|
| [6] | Y. Long, X. M. Yuan, T. S. Du, Simpson-like inequalities for functions whose third derivatives belong to $s$-convexity involoving Atangana-Baleanu fractional integrals and their applications, Filomat, 38 (2024), 9373–9397. |
| [7] |
F. Jarad, E. Ugurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 247. https://doi.org/10.1186/s13662-017-1306-z doi: 10.1186/s13662-017-1306-z
|
| [8] | A. M. Mercer, A variant of Jensen's inequality, J. Inequal. Pure Appl. Math., 4 (2003), 73. |
| [9] | M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. |
| [10] | A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and applications to heat tranfer model, Thermal Sci., 20 (2016), 763–769. |
| [11] |
M. Tariq, H. Ahmad, S. K. Sahoo, A. Kashuri, T. A. Nofal, C. H. Hsu, Inequalities of Simpson-Mercer-type including Atangana-Baleanu fractional operators and their applications, AIMS Math., 7 (2022), 15159–15181. https://doi.org/10.3934/math.2022831 doi: 10.3934/math.2022831
|
| [12] |
S. Kermausuor, E. R. Nwaeze, New fractional integral inequalities via $k$-Atangana-Baleanu fractional integral operators, Fractal Fract., 7 (2023), 740. https://doi.org/10.3390/fractalfract7100740 doi: 10.3390/fractalfract7100740
|
Jen Chieh Lo. Some Hermite-Hadamard-Mercer type inequalities via Atangana-Baleanu-conformable fractional operator[J]. AIMS Mathematics, 2026, 11(6): 17937-17950. doi: 10.3934/math.2026731
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