Fractional Hermite-Hadamard Inequalities for Different Convex Classes Based on $\psi$-Hilfer-Atangana-Baleanu Operators

Year 2025, Volume: 11 Issue: 1-2, 32 - 38, 29.12.2025

Barış Çelik , Erhan Set

Abstract

Hermite-Hadamard type inequalities play a central role in the study of convexity and its generalizations, providing a fundamental tool for both theoretical analysis and applications. In recent years, fractional integral operators have been widely employed to establish new versions of these inequalities. Among them, the $\psi$-Hilfer-Atangana-Baleanu ($\psi$-HAB) fractional integral operators have attracted attention as a powerful extension of the Atangana-Baleanu and ABK operators, offering a flexible framework to explore generalized convexity classes.

In this paper, with the help of the identities proved by Kermausor et al. in \cite{Kermausuor2025}, we obtain some new integral inequalities of Hermite-Hadamard type for the quasi-convex function and the $P$-function, respectively.

Keywords

Hermite-Hadamard inequalities , quasi-convex function , $P$-function , $\psi$-Hilfer-Atangana-Baleanu fractional integral operators

References

• AKDEMİR, A.O., KARAOĞLAN, A., RAGUSA, M.A., SET, E. (2021). Fractional Integral Inequalities via Atangana-Baleanu Operators for Convex and Concave Functions, J. Funct. Spaces, 2021(1), Article ID 1055434, 1–10.
• ATANGANA, A., BALEANU, D. (2016). New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20(2), 763–769.
• AVCI-ARDIÇ, M., AKDEMİR, A.O., KAVURMACI, H. (2023). Integral inequalities for differentiable s-convex functions in the second sense via Atangana-Baleanu fractional integral operators, Filomat, 37(18), 6229–6244.
• AZZOUZ, N., BENAISSA, B., BUDAK, H., DEMİR, İ. (2025). Hermite-Hadamard-Mercer type inequalities for fractional integrals: A study with h-convexity and ψ-Hilfer operators, Bound Value Probl., 2025, Article number 14.
• BUDAK, H., ALI, M.A., KASHURI, A. (2022). New inequalities of Hermite-Hadamard type for twice differentiable functions via generalized fractional integrals, Turkish J. Ineq., 6(2), 27–39.
• BUTT, S.I., KHAN, D., JAIN, S., OROS, G.I., AGARWAL, P., MOMANI, S. (2025). Fractional integral inequalities for Superquadratic functions via Atangana Baleanu’s operator with applications, Fractals, 33(8), 1–24.
• COŞKUN, M., YILDIZ, Ç., COTIRLA, L.-I. (2024). Novel Estimations of Hadamard-Type Integral Inequalities for Raina’s Fractional Operators, Fractal and Fractional, 8(5), 302.
• ÇELİK, B., ÖZDEMİR, M.E., AKDEMİR, A.O., SET, E. (2021). Integral Inequalities for some convexity classes via Atangana-Baleanu Integral Operators, Turkish J. Ineq., 5(2), 82–92.
• HEZENCİ, F., KARA, H., BUDAK, H. (2023). Conformable fractional versions of Hermite-Hadamard-type inequalities for twice-differentiable functions, Bound. Value Probl., 2023(1), Article ID 48.
• ION, D.A. (2007). Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of University of Craiova, Math. Comp. Sci. Ser., 34, 82–87.
• KERMAUSUOR, S., DENU, D., Olanipekun, P.O. (2026). ψ-Hilfer-Atangana-Baleanu fractional integral operators and some fractional integral inequalities, Kragujevac J. Math., 50(10), 1585–1604.
• KILBAS, A.A., SRIVASTAVA, H.M., TRUJILLO, J. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006.
• MUNIR, A., KASHURI, A., HEZENCI, F., KARA, H. (2025). Novelty Hermite-Hadamard-type inequalities for various convexity functions and their applications, Sahand Commun. Math. Anal., 2(4), 213–239.
• PACHPATTE, B.G. (2005). Mathematical Inequalities, Elsevier, Amsterdam.
• PEARCE, C.E.M., RUBINOV, A.M. (1999). P-functions, quasi-convex functions and Hadamard-type inequalities, J. Math. Anal. Appl., 240(1), 92–104.
• PECARIC, J., PROSCHAN, F., TONG, Y.L. (1992). Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston.
• SARIKAYA, M.Z., SET, E., YALDIZ, H., BAŞAK, N. (2013). Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57, 2403–2407.
• SET, E., ÇELİK, B. (2018). Generalized fractional Hermite-Hadamard type inequalities for convex and convex functions, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 67(1), 333–344.
• SET E., CHOI, J., ÇELİK, B. (2018). New Hermite-Hadamard type inequalities for product of different convex functions involving certain fractional integral operators, J. Math. Comp. Sci., 18(1), 29–36.
• XI, B.-Y., BAI, R.F., QI, F. (2012). Hermite-Hadamard type inequalities for the m- and (α,m)-geometrically convex functions, Aequationes Math., 84(3), 261–269.
• YALÇIN, A., GÜL, E., AKDEMİR, A.O. (2025). Hermite-Hadamard type inequalities for Co-ordinated convex functions with variable-order fractional integrals, Appl. Comput. Math., 24(2), 326–343.