Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals with Applications

Arslan Munir; Hüseyin Budak; Artion Kashuri

doi:10.32323/ujma.1720774

Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals with Applications

Abstract

Fractional integral inequalities have emerged as powerful and versatile tools in advancing both pure and applied mathematics in recent years. Numerous researchers have recently introduced various generalized inequalities involving fractional integral operators. In this paper, we develop new Hermite-Hadamard-Mercer type inequalities that extend the classical Hermite-Hadamard framework to the setting of the Riemann-Liouville fractional integral. Additionally, we establish Hermite-Hadamard-Mercer type inequalities for functions whose absolute second derivatives are convex. Moreover, we present a new midpoint-type inequality based on the well-known power-mean inequality. Several applications are also provided, including new results related to special means, the midpoint formula, quadrature formulas, the $q$-digamma function, and modified Bessel functions.

Keywords

Convex function, Error estimation, Hermite-Hadamard Mercer-type inequality, Modified Bessel function, Power-mean inequality, $q$-Digamma function, Riemann-Liouville fractional integrals, Special means

References

[1] Y. Qin, Integral and Discrete Inequalities and Their Applications, Vol. 1, Birkhäuser, Berlin, 2016. https://doi.org/10.1007/978-3-319-33301-4
[2] A. Praveen, S. S. Dragomir, M. Jleli, B. Samet, Advances in Mathematical Inequalities and Applications, Springer, Singapore, 2018. https://doi.org/10.1007/978-981-13-3013-1
[3] N. Mehmood, S. I. Butt, D. Pečarić, J. Pečarić, Generalizations of cyclic refinements of Jensen’s inequality by Lidstone’s polynomial with applications in information theory, J. Math. Inequal., 14(1) (2019), 249–271. https://doi.org/10.7153/jmi-2020-14-17
[4] T. S. Du, M. U. Awan, A. Kashuri, S. Zhao, Some k-fractional extensions of the trapezium inequalities through generalized relative semi-(m,h)-preinvexity, Appl. Anal., 100(3) (2021), 642–662. https://doi.org/10.1080/00036811.2019.1616083
[5] T. Du, H. Wang, M. A. Khan, Y. Zhang, Certain integral inequalities considering generalized m-convexity on fractal sets and their applications, Fractals, 27(7) (2019), Article ID 1950117. https://doi.org/10.1142/S0218348X19501172
[6] G. A. Anastassiou, Advances Inequalities, Vol. 11, World Scientific, 2010. https://doi.org/10.1142/7847
[7] N. Azzouz, B. Benaissa, Exploring Hermite–Hadamard-type inequalities via $\psi$-conformable fractional integral operators, J. Inequal. Math. Anal., 1(1) (2025), 15–27. https://doi.org/10.63286/jima.2025.02
[8] S. Qaisar, A. Munir, H. Budak, Certain fractional inequalities via the Caputo–Fabrizio operator, Filomat, 37(29) (2023), 10093–10106. https://doi.org/10.2298/FIL2329093Q
[9] D. A. Ion, Some estimates on the Hermite–Hadamard inequality through quasi-convex functions, Ann. Univ. Craiova Math. Comp. Sci. Ser., 34 (2007), 82–87.
[10] S. Qaisar, A. Munir, M. Naeem, H. Budak, Some Caputo–Fabrizio fractional integral inequalities with applications, Filomat, 38(16) (2024), 5905–5923. https://doi.org/10.2298/FIL2416905Q

[11] M. E. Özdemir, M. Avcı, E. Set, On some inequalities of Hermite–Hadamard type via m-convexity, Appl. Math. Lett., 23(9) (2010), 1065–1070. https://doi.org/10.1016/j.aml.2010.04.037
[12] J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier d’une fonction consid´er´ee par Riemann, J. Math. Pures Appl., 9 (1893), 171–216.
[13] C. Hermite, Sur deux limites dune integrale definie, Mathesis, 3 (1883), Article ID 182.
[14] A. M. Mercer, A variant of Jensen’s inequality, J. Inequal. Pure Appl. Math., 4(4) (2003), Article ID 73.
[15] H. R. Moradi, S. Furuichi, Improvement and generalization of some Jensen–Mercer-type inequalities, J. Math. Inequal., 14(2) (2020), 377–383. https://doi.org/10.7153/jmi-2020-14-24
[16] M. A. Khan, Z. Husain, Y. M. Chu, New estimates for Csiszár divergence and Zipf–Mandelbrot entropy via Jensen–Mercer’s inequality, Complexity, 2020(1) (2020), Article ID 8928691. https://doi.org/10.1155/2020/8928691
[17] M. Vivas-Cortez, M. Z. Javed, M. U. Awan, M. A. Noor, S. S. Dragomir, Bullen–Mercer type inequalities with applications in numerical analysis, Alexandria Eng. J., 96 (2024), 15–33. https://doi.org/10.1016/j.aej.2024.03.093
[18] M. Vivas-Cortez, M. U. Awan, U. Asif, M. Z. Javed, H. Budak, Advances in Ostrowski–Mercer like inequalities within Fractal space, Fractal Fract., 7(9) (2023), Article ID 689. https://doi.org/10.3390/fractalfract7090689
[19] B. Bin-Mohsin, M. Z. Javed, M. U. Awan, M. V. Mihai, H. Budak, A. G. Khan, M. A. Noor, Jensen–Mercer type inequalities in the setting of fractional calculus with applications, Symmetry, 14(10) (2022), Article ID 2187. https://doi.org/10.3390/sym14102187
[20] K. Nonlaopon, M. U. Awan, U. Asif, M. Z. Javed, I. Slimane, A. Kashuri, Fractional Jensen–Mercer type inequalities involving generalized Raina’s function and applications, Symmetry, 14(10) (2022), Article ID 2204. https://doi.org/10.3390/sym14102204
[21] M. Kian, M. S. Moslehian, Refinements of the operator Jensen–Mercer inequality, Electron. J. Linear Algebra, 26 (2013), 742–753.
[22] H. Öğülmüş, M. Z. Sarıkaya, Hermite–Hadamard–Mercer type inequalities for fractional integrals, Filomat, 35(7) (2021), 2425–2436. https://doi.org/10.2298/FIL2107425O+
[23] Ç. Yildiz, T. İşleyen, L. I. Cotirla, New results on majorized discrete Jensen–Mercer inequality for Raina fractional operators, Fractal Fract., 9(6) (2025), Article ID 343. https://doi.org/10.3390/fractalfract9060343
[24] Ç. Yildiz, S. Erden, S. Kermausuor, D. Breaz, L. I. Cotirla, New estimates on generalized Hermite–Hadamard–Mercer-type inequalities, Bound. Value Probl., 2025 (2025), Article ID 19. https://doi.org/10.1186/s13661-025-02012-y
[25] H. H. Chu, S. Rashid, Z. Hammouch, Y. M. Chu, New fractional estimates for Hermite–Hadamard–Mercer’s type inequalities, Alexandria Eng. J., 59(5) (2020), 3079–3089. https://doi.org/10.1016/j.aej.2020.06.040
[26] Z. Çiftci, M. Coşkun, Ç. Yildiz, L. I. Cotirla, D. Breaz, On new generalized Hermite–Hadamard–Mercer-type inequalities for Raina functions, Fractal Fract., 8(8) (2024), Article ID 472. https://doi.org/10.3390/fractalfract8080472
[27] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198, Elsevier, 1999.
[28] R. Gorenflo, F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, Springer, Vienna, 1997. https://doi.org/10.48550/arXiv.0805.3823
[29] M. Z. Sarikaya, A. Saglam, H. Yildirim, New inequalities of Hermite Hadamard type for functions whose second derivatives absolute values are convex and quasi convex, Int. J. Open Probl. Comput. Sci. Math., 5(3) (2012), 1–14.
[30] G. N. Watson, A Treatise on the Theory of Bessel Functions, The University Press, 1922.

Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Arslan Munir ^*
0009-0008-8148-5607
China

Hüseyin Budak
0000-0001-8843-955X
Türkiye

Artion Kashuri
0000-0003-0115-3079
Albania

Early Pub Date

November 27, 2025

Publication Date

December 11, 2025

Submission Date

June 16, 2025

Acceptance Date

October 4, 2025

Published in Issue

Year 2025 Volume: 8 Number: 4

DOI

https://doi.org/10.32323/ujma.1720774

IZ

https://izlik.org/JA79NT29HW

APA

Munir, A., Budak, H., & Kashuri, A. (2025). Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals with Applications. Universal Journal of Mathematics and Applications, 8(4), 167-178. https://doi.org/10.32323/ujma.1720774

AMA

1.Munir A, Budak H, Kashuri A. Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals with Applications. Univ. J. Math. Appl. 2025;8(4):167-178. doi:10.32323/ujma.1720774

Chicago

Munir, Arslan, Hüseyin Budak, and Artion Kashuri. 2025. “Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals With Applications”. Universal Journal of Mathematics and Applications 8 (4): 167-78. https://doi.org/10.32323/ujma.1720774.

EndNote

Munir A, Budak H, Kashuri A (December 1, 2025) Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals with Applications. Universal Journal of Mathematics and Applications 8 4 167–178.

IEEE

[1]A. Munir, H. Budak, and A. Kashuri, “Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals with Applications”, Univ. J. Math. Appl., vol. 8, no. 4, pp. 167–178, Dec. 2025, doi: 10.32323/ujma.1720774.

ISNAD

Munir, Arslan - Budak, Hüseyin - Kashuri, Artion. “Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals With Applications”. Universal Journal of Mathematics and Applications 8/4 (December 1, 2025): 167-178. https://doi.org/10.32323/ujma.1720774.

JAMA

1.Munir A, Budak H, Kashuri A. Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals with Applications. Univ. J. Math. Appl. 2025;8:167–178.

MLA

Munir, Arslan, et al. “Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals With Applications”. Universal Journal of Mathematics and Applications, vol. 8, no. 4, Dec. 2025, pp. 167-78, doi:10.32323/ujma.1720774.

Vancouver

1.Arslan Munir, Hüseyin Budak, Artion Kashuri. Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals with Applications. Univ. J. Math. Appl. 2025 Dec. 1;8(4):167-78. doi:10.32323/ujma.1720774