Significant Approach of Hermite-Hadamard Type Fractional Inequalities for MET-\((p,s)\)-Convex Function and its Applications

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DOI: https://doi.org/10.28924/2291-8639-24-2026-229
Published: Aug 3, 2026
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Mamoona Siddiq, Fazilat Bibi, Rana Safder, Jamal Salah, Aseel Smerat, Significant Approach of Hermite-Hadamard Type Fractional Inequalities for MET-\((p,s)\)-Convex Function and its Applications, Int. J. Anal. Appl., 24 (2026), 229.

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Mamoona Siddiq, Fazilat Bibi, Rana Safder, Jamal Salah, Aseel Smerat

Abstract

There is a very close relation between convexity and fractional operators to improve well-known inequalities. Different types of inequality have been examined, providing new ways in the inequality literature. In this paper, we discuss Hermite-Hadamard extensions by successfully implementation of modified exponential trigonometry (MET)-\((p,s)\)-convexity with Riemann–Liouvile fractional operator (RL-FO). Moreover, we extract some corollaries which show the straingth of our main results. Furthermore, we discuss numerical implementation for specfic cases, special values of the parameters, and demonstrate that the derived outcome generalizes numerous inequalities that have been previously found in the literature.

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References

  1. O. Almutairi, A. Kilicman, New Fractional Inequalities of Midpoint Type via s-Convexity and Their Application, J. Inequal. Appl. 2019 (2019), 267. https://doi.org/10.1186/s13660-019-2215-3.
  2. R. Ali, A. Mukheimer, T. Abdeljawad, S. Mubeen, S. Ali, et al., Some New Harmonically Convex Function Type Generalized Fractional Integral Inequalities, Fractal Fract. 5 (2021), 54. https://doi.org/10.3390/fractalfract5020054.
  3. K.D. Budak, M.U. Gurbuz, Modified Exponential Trigonometric Convex Functions and Some Integral Inequalities, Turk. J. Inequal. 8 (2024), 60-72.
  4. M. Caputo, M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Prog. Fract. Differ. Appl. 1 (2015), 73-85.
  5. F. Chen, Extensions of the Hermite-Hadamard Inequality for Convex Functions via Fractional Integrals, J. Math. Inequal. 10 (2016), 75-81. https://doi.org/10.7153/jmi-10-07.
  6. H. Chen, U.N. Katugampola, Hermite–Hadamard and Hermite–Hadamard–Fejér Type Inequalities for Generalized Fractional Integrals, J. Math. Anal. Appl. 446 (2017), 1274-1291. https://doi.org/10.1016/j.jmaa.2016.09.018.
  7. S.S. Dragomir, Two Mappings in Connection to Hadamard's Inequalities, J. Math. Anal. Appl. 167 (1992), 49-56. https://doi.org/10.1016/0022-247X(92)90233-4.
  8. S.S. Dragomir, R.P. Agarwal, Two Inequalities for Differentiable Mappings and Applications to Special Means of Real Numbers and to Trapezoidal Formula, Appl. Math. Lett. 11 (1998), 91-95. https://doi.org/10.1016/S0893-9659(98)00086-X.
  9. S.S. Dragomir, S. Fitzpatrick, The Hadamard Inequalities for s-Convex Functions in the Second Sense, Demonstr. Math. 32 (1999), 687-696. https://doi.org/10.1515/dema-1999-0403.
  10. L. Fejer, Uber die Fourierreihen II, Mat. Természettud. Ert. 24 (1906), 369-390.
  11. M. Eshaghi Gordji, M. Rostamian Delavar, M. De La Sen, On φ-Convex Functions, J. Math. Inequal. 10 (2016), 173-183. https://doi.org/10.7153/jmi-10-15.
  12. A. Guessab, Sharp Approximations Based on Delaunay Triangulations and Voronoi Diagrams, NSU Publishing and Printing Center, 2022.
  13. J.C. Huang, G. Rahman, K.S. Nisar, A. Ghaffar, F. Qi, Some Inequalities of Hermite-Hadamard Type for k-Fractional Conformable Integrals, Aust. J. Math. Anal. Appl. 16 (2019), 1-9.
  14. H. Hudzik, L. Maligranda, Some Remarks on s-Convex Functions, Aequ. Math. 48 (1994), 100-111. https://doi.org/10.1007/BF01837981.
  15. I. Iscan, Ostrowski Type Inequalities for p-Convex Functions, New Trends Math. Sci. 4 (2016), 140-150. https://doi.org/10.20852/ntmsci.2016318838.
  16. I. İşcan, S. Wu, Hermite–Hadamard Type Inequalities for Harmonically Convex Functions via Fractional Integrals, Appl. Math. Comput. 238 (2014), 237-244. https://doi.org/10.1016/j.amc.2014.04.020.
  17. H. Kadakal, Hermite-Hadamard Type Inequalities for Trigonometrically Convex Functions, Sci. Stud. Res. Ser. Math. Inform. 28 (2018), 19-28.
  18. M. Kadakal, I. Iscan, P. Agarwal, M. Jleli, Exponential Trigonometric Convex Functions and Hermite-Hadamard Type Inequalities, Math. Slovaca 71 (2021), 43-56. https://doi.org/10.1515/ms-2017-0410.
  19. M.B. Khan, S. Treanta, H. Budak, Generalized p-Convex Fuzzy-Interval-Valued Functions and Inequalities Based upon the Fuzzy-Order Relation, Fractal Fract. 6 (2022), 63. https://doi.org/10.3390/fractalfract6020063.
  20. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
  21. K.S. Nisar, G. Rahman, K. Mehrez, Chebyshev Type Inequalities via Generalized Fractional Conformable Integrals, J. Inequal. Appl. 2019 (2019), 245. https://doi.org/10.1186/s13660-019-2197-1.
  22. K.S. Nisar, A. Tassaddiq, G. Rahman, A. Khan, Some Inequalities via Fractional Conformable Integral Operators, J. Inequal. Appl. 2019 (2019), 217. https://doi.org/10.1186/s13660-019-2170-z.
  23. J.E. Pecaric, F. Proschan, Y.L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, 1992.
  24. I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
  25. F. Qi, G. Rahman, S. Hussain, W.-S. Du, K. Nisar, Some Inequalities of Čebyšev Type for Conformable k-Fractional Integral Operators, Symmetry 10 (2018), 614. https://doi.org/10.3390/sym10110614.
  26. X. Qiang, G. Farid, M. Yussouf, K.A. Khan, A.U. Rahman, New Generalized Fractional Versions of Hadamard and Fejér Inequalities for Harmonically Convex Functions, J. Inequal. Appl. 2020 (2020), 191. https://doi.org/10.1186/s13660-020-02457-y.
  27. G. Rahman, T. Abdeljawad, F. Jarad, K.S. Nisar, Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications, Mathematics 8 (2020), 113. https://doi.org/10.3390/math8010113.
  28. G. Rahman, K.S. Nisar, F. Qi, Some New Inequalities of the Grüss Type for Conformable Fractional Integrals, AIMS Math. 3 (2018), 575-583. https://doi.org/10.3934/Math.2018.4.575.
  29. H.H. Toader, Some Generalizations of the Convexity, in: Proceedings of the Colloquium on Approximation and Optimization, Cluj-Napoca, Romania, 1984, pp. 329-338.
  30. M. Rostamian Delavar, S. Mohammadi Aslani, M. De La Sen, Hermite-Hadamard-Fejér Inequality Related to Generalized Convex Functions via Fractional Integrals, J. Math. 2018 (2018), 1-10. https://doi.org/10.1155/2018/5864091.
  31. E.K. Godunova, V.I. Levin, Inequalities for Functions of a Broad Class That Contains Convex, Monotone and Some Other Forms of Functions, in: Proceedings of Numerical Mathematics and Mathematical Physics, Moskovskii Gosudarstvennyi Pedagogicheskii Institut, pp. 138-142, (1985).
  32. M.S. Zahoor, A. Hussain, Y. Wang, New Fractional Hermite–Hadamard-Type Inequalities for Caputo Derivative and MET-(p, s)-Convex Functions with Applications, Fractal Fract. 10 (2026), 62. https://doi.org/10.3390/fractalfract10010062.
  33. M.V. Mihai, M.U. Awan, M.A. Noor, J.K. Kim, K.I. Noor, Hermite–Hadamard Inequalities and Their Applications, J. Inequal. Appl. 2018 (2018), 309. https://doi.org/10.1186/s13660-018-1895-4.
  34. M. Muddassar, M.I. Bhatti, W. Irshad, Generalisations of Integral Inequalities of Hermite–Hadamard Type through Convexity, Bull. Aust. Math. Soc. 88 (2012), 320-330. https://doi.org/10.1017/S0004972712000937.