Some Generalized Fractional Hermite-Hadamard-Type Inequalities for m−Convex Functions
DOI:
https://doi.org/10.31489/2025m3/85-96Keywords:
integral inequality, fractional Hermite-Hadamard inequality, convex functions, m−convex functions, twice differentiable functions, Euler Beta function, Hö¨lder’s integral inequality, power mean integral inequalityAbstract
Fractional Hermite-Hadamard-type inequalities represent a significant area of study in convex analysis due to their extensive applications in mathematical and applied sciences. These inequalities provide powerful tools for estimating the integral mean of a convex function in terms of its values at the endpoints of a given interval. In this paper, we focus on the development and refinement of fractional Hermite-Hadamardtype inequalities for the class of twice differentiable m-convex functions. Utilizing advanced analytical techniques, such as Ho¨lder’s inequality and the power mean integral inequality, we derive new bounds that generalize existing results in the literature. These findings not only extend classical inequalities to a broader class of convex functions but also provide sharper and more versatile estimations. The presented results are expected to have significant implications in various mathematical domains, including fractional calculus, optimization, and mathematical modeling. This work contributes to the ongoing efforts to generalize and refine integral inequalities by incorporating fractional operators and broader convexity assumptions, offering a deeper understanding of the behavior of m-convex functions under fractional integration.
References
Set, E., Ozdemir, M.E., Sarikaya, M.Z., & Karakoc, F. (2017). Hermite–Hadamard-type inequal-¨ ities for (α, m)−convex functions via fractional integrals. Moroccan J. Pure and Appl. Anal., 3(1), 15–21. https://doi.org/10.1515/mjpaa-2017-0002
Bilal, M., & Khan, A.R. (2021). On Some New Hermite–Hadamard Dual Inequalities. JMCMS, 16(7), 93–98. https://doi.org/10.26782/jmcms.2021.07.00008
Bilal, M., & Khan, A.R. (2021). New Generalized Hermite–Hadamard Inequalities for p-convex functions in the mixed kind. EJPAM, 14(3), 863–880. https://doi.org/10.29020/nybg.ejpam.v14i3.4015
Bilal, M., & Khan, A.R. (2024). Refinement of Hermite–Hadamard-type inequalities for convex functions with applications. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., 31(1), 33–48. https://doi.org/10.7468/jksmeb.2024.31.1.33
Lara, T., Quintero, R., & Rosales, E. (2017). m-convexity and functional equations. Moroccan J. Pure and Appl. Anal., 3(1), 56–62. https://doi.org/10.1515/mjpaa-2017-0005
Toader, G. (1984). Some generalizations of the convexity. Proc. Colloq. Aprox. Optim., 329–338.
Alomari, M., & Darus, M. (2008). Hadamard-type inequalities for s-convex functions. Int. Math. Forum, 3(40), 1965–1975.
Lakhdari, A., Budak, H., Mlaiki, N., Meftah, B., & Abdeljawad, T. (2025). New insights on fractal–fractional integral inequalities: Hermite–Hadamard and Milne estimates. Chaos, Solitons and Fractals, 193, 116087. https://doi.org/10.1016/j.chaos.2025.116087
Lakhdari, A., Mohsin, B.B., Jarad, F., Xu, H., & Meftah, B. (2024). A parametrized approach to generalized fractional integral inequalities: Hermite–Hadamard and Maclaurin variants. Journal of King Saud University - Science, 36(11), 103523. https://doi.org/10.1016/j.jksus.2024.103523
Li, H., Meftah, B., Saleh, W., Xu, H., Kili¸cman, A., & Lakhdari, A. (2024). Further Hermite– Hadamard-type inequalities for fractional integrals with exponential kernels. Fractal and Fractional, 8(6), 345. https://doi.org/10.3390/fractalfract8060345
Bayraktar, B., & Kudaev, V.C. (2019). Some new integral inequalities for (s,m)-convex and (α,m)-convex functions. Bulletin of the Karaganda University. Mathematics Series, 2(94), 15–25. https://doi.org/10.31489/2019m2/15-25
Ozcan, S. (2020). Hermite–Hadamard-type inequalities for¨ m-convex and (α,m)-convex functions. J. Inequal. Appl., 2020, Article 175. https://doi.org/10.1186/s13660-020-02442-5
Khan, A.R., Khan, I.U., & Muhammad, S. (2021). Hermite–Hadamard-type fractional integral inequalities for s-convex functions of mixed kinds. TMCS, 1(1), 25–37.
Ozcan, S., & I¸scan, I. (2019). Some new Hermite–Hadamard-type inequalities for¨ s-convex functions and their applications. Journal of Inequalities and Applications, 2019, 201. https://doi.org/10.1186/s13660-019-2151-2
Noor, M.A., & Awan, M.U. (2013). Some integral inequalities for two kinds of convexities via fractional integrals. TJMM, 5(2), 129–136.
Dragomir, S.S., Bhatti, M.I., Iqbal, M., & Muddassar, M. (2015). Some new Hermite–Hadamard’s type fractional integral inequalities. JCAA, 18(4), 655–661.
Ozdemir, M.E., Avci, M., & Set, E. (2010). On some inequalities of Hermite–Hadamard-type via¨ m-convexity. Appl. Math. Lett., 23(9), 1065–1070. https://doi.org/10.1016/j.aml.2010.04.037
Hussain, M., Bhatti, M.I., & Iqbal, M. (2009). Hadamard-type inequalities for s-convex functions I. Punjab University Journal of Mathematics, 41, 51–60.
