Some New Notions of Mathematical Integral Inequalities: Theory and Applications

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DOI: https://doi.org/10.28924/2291-8639-24-2026-175
Published: May 29, 2026
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Hijaz Ahmad, Muhammad Tariq, Ali Asghar, Waqar Afzal, Maggie Aphane, Rakib Efendiev, Some New Notions of Mathematical Integral Inequalities: Theory and Applications, Int. J. Anal. Appl., 24 (2026), 175.

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Hijaz Ahmad, Muhammad Tariq, Ali Asghar, Waqar Afzal, Maggie Aphane, Rakib Efendiev

Abstract

Convex analysis and mathematical inequalities play a fundamental role in both pure and applied sciences. In this work, we first explore the notion of n-fractional polynomial s-like m-convexity involving Raina’s mapping and also its algebraic properties. We then introduce a novel Hermite–Hadamard (H-H), midpoint H-H, trapezoid H-H type inequalities based on this generalized concept and the k-fractional operator. Several related corollaries and examples are examined, particularly in connection with the Mittag–Leffler function. The practical utility of the proposed inequalities is demonstrated through applications to viscoelastic materials with fractional damping, supported by computational algorithms, and a numerical example involving fractional diffusion in fractured media. The results provide meaningful refinements and novel insights that extend and enrich existing research in the field.

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References

  1. S. Boyd, C. Crusius, A. Hansson, New Advances in Convex Optimization and Control Applications, IFAC Proc. Vol. 30 (1997), 365–393. https://doi.org/10.1016/S1474-6670(17)43183-1.
  2. T. Pennanen, Convex Duality in Stochastic Optimization and Mathematical Finance, Math. Oper. Res. 36 (2011), 340–362. https://doi.org/10.1287/moor.1110.0485.
  3. Z.Q. Luo, W. Yu, An Introduction to Convex Optimization for Communications and Signal Processing, IEEE J. Sel. Areas Commun. 24 (2006), 1426–1438. https://doi.org/10.1109/JSAC.2006.879347.
  4. J. Green, W.P. Heller, Chapter 1 Mathematical Analysis and Convexity with Applications to Economics, in: Handbook of Mathematical Economics, Elsevier, 1981: pp. 15–52. https://doi.org/10.1016/s1573-4382(81)01005-9.
  5. H. Föllmer, A. Schied, Convex Measures of Risk and Trading Constraints, Financ. Stochastics 6 (2002), 429–447. https://doi.org/10.1007/s007800200072.
  6. J. Pełczyński, Application of the Theory of Convex Sets for Engineering Structures with Uncertain Parameters, Appl. Sci. 10 (2020), 6864. https://doi.org/10.3390/app10196864.
  7. M. Tariq, H. Ahmad, S.K. Sahoo, A. Kashuri, T.A. Nofal, et al., 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.
  8. R. Bapat, Applications of an Inequality in Information Theory to Matrices, Linear Algebr. Appl. 78 (1986), 107–117. https://doi.org/10.1016/0024-3795(86)90018-2.
  9. M.J. Cloud, B.C. Drachman, L.P. Lebedev, Inequalities, Springer, 2014. https://doi.org/10.1007/978-3-319-05311-0.
  10. M. A. Latif, On Hermite–Hadamard Type Integral Inequalities for $n$-Times Differentiable Preinvex Functions with Applications, Stud. Univ. Babec{s}-Bolyai Math. 58 (2013), 325–343.
  11. S. Özcan, Some Integral Inequalities of Hermite-Hadamard Type for Multiplicatively Preinvex Functions, AIMS Math. 5 (2020), 1505–1518. https://doi.org/10.3934/math.2020103.
  12. A. Iqbal, M.A. Khan, N. Mohammad, E.R. Nwaeze, Y.M. Chu, Revisiting the Hermite-Hadamard Fractional Integral Inequality via a Green Function, AIMS Math. 5 (2020), 6087–6107. https://doi.org/10.3934/math.2020391.
  13. S.K. Sahoo, P.O. Mohammed, B. Kodamasingh, M. Tariq, Y.S. Hamed, New Fractional Integral Inequalities for Convex Functions Pertaining to Caputo–Fabrizio Operator, Fractal Fract. 6 (2022), 171. https://doi.org/10.3390/fractalfract6030171.
  14. M. Tariq, H. Ahmad, A.G. Shaikh, S.K. Sahoo, et al., New Fractional Integral Inequalities for Preinvex Functions Involving Caputo-Fabrizio Operator, AIMS Math. 7 (2022), 3440–3455. https://doi.org/10.3934/math.2022191.
  15. M. Tariq, Hermite-Hadamard Type Inequalities via $p$–Harmonic Exponential Type Convexity and Applications, Univ. J. Math. Appl. 4 (2021), 59–69. https://doi.org/10.32323/ujma.870050.
  16. S.I. Butt, M. Umar, K.A. Khan, A. Kashuri, H. Emadifar, Fractional Hermite–Jensen–Mercer Integral Inequalities with Respect to Another Function and Application, Complexity 2021 (2021), 9260828. https://doi.org/10.1155/2021/9260828.
  17. O. Moaaz, I. Dassios, W. Muhsin, A. Muhib, Oscillation Theory for Non-Linear Neutral Delay Differential Equations of Third Order, Appl. Sci. 10 (2020), 4855. https://doi.org/10.3390/app10144855.
  18. S.I. Butt, H. Budak, M. Tariq, M. Nadeem, Integral Inequalities for n‐Polynomial s‐Type Preinvex Functions with Applications, Math. Methods Appl. Sci. 44 (2021), 11006–11021. https://doi.org/10.1002/mma.7465.
  19. M. Tariq, S.K. Ntouyas, A.A. Shaikh, A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Fractional Integral Operators, Mathematics 11 (2023), 1953. https://doi.org/10.3390/math11081953.
  20. H. Ahmad, M. Nadeem, A. Asghar, R. Efendiev, M. Tariq, et al., Some New Approaches of Hermite-Hadamard Type Inequalities Pertaining to Generalized Convexity on Coordinates Using Hypergeometric Function, Azerbaijan J. Math. (2026), 220. https://doi.org/10.59849/2218-6816.2026.1.220.
  21. S. Khan, N.A. Shah, H. Ahmad, A. Kamel, W.M. Abdelfattah, et al., Partial Sums for Normalized Mittag-Leffler-Prabhakar Function and Barnes-Mittag-Leffler Function, Eur. J. Pure Appl. Math. 18 (2025), 7027. https://doi.org/10.29020/nybg.ejpam.v18i4.7027.
  22. M. Zafarullah, I. Khan, A. Nawaz, M. Jamil, T.N. Alharthi, et al., Implementation of Atangana-Baleanu-Caputo (ABC) Fractional Time Operator on Heat and Mass Transfer Phenomena of Walter's-B Fluid, Fractals (2026), 2640051. https://doi.org/10.1142/s0218348x26400517.
  23. P. Karthiga, S.M. Sivalingam, V. Govindaraj, H. Ahmad, Controllability Analysis of Fractional Nonlinear Dynamical Systems Using $Psi$-Caputo Derivatives and Prescribed Controls, J. Taibah Univ. Sci. 20 (2026), 2635196. https://doi.org/10.1080/16583655.2026.2635196.
  24. N.A. Sheikh, Z.B. Ismail, I. Khan, Influence of Memory Effects on Heat and Mass Transfer in Fractional Casson–Brinkman Electrically Conducting Flow with Ramped Boundaries, Fractals (2026), 2650073. https://doi.org/10.1142/S0218348X26500738.
  25. F. Faisal, Z. Haider, U. Ghani, H. Ali, M. Amjad, et al., Derivation of Hermite–Hadamard–Type Inequalities via Quasi-Preinvex Functions and Strongly Preinvex Functions, Int. J. Geom. Methods Mod. Phys. (2025), 2650048. https://doi.org/10.1142/S0219887826500489.
  26. N. Khan, F. Ali, G. Alhamzi, I. Khan, M. Bakouri, et al., Heat Transfer Enhancement in MHD Flow of Tri-Hybrid Maxwell Nanofluid with Ramped Wall Heating: A Fractional Caputo–Crank–Nicolson Approach, Results Eng. 29 (2026), 109476. https://doi.org/10.1016/j.rineng.2026.109476.
  27. M. Lavanya, B.S. Vadivoo, H. Ahmad, D.U. Ozsahin, T. Radwan, Langevin Neutral Impulsive Fractional Stochastic System Along Fractional Brownian Motion-A Controllability Analysis, J. Low Freq. Noise, Vib. Act. Control. (2026), 14613484261430389. https://doi.org/10.1177/14613484261430389.
  28. A. Rayal, J. Kaur, P.A. Patel, D.U. Ozsahin, H. Ahmad, et al., A Spectral Collocation Scheme with 2D Ultraspherical Wavelets for Fractional Nonlinear Gas Dynamics Equations Under Caputo–Fabrizio Derivative, Fractals (2026), 2640019. https://doi.org/10.1142/S0218348X26400190.
  29. R. Balamurugan, I. Khan, M. Bakouri, T. Alqahtani, Physics-Informed Neural Network Approach to Unsteady Fractional Flow in a Vertical Coaxial Annulus with Thermal Effects and Magneto-Hall Interaction, Results Eng. 29 (2026), 109965. https://doi.org/10.1016/j.rineng.2026.109965.
  30. S.I. Butt, M. Umar, S. Rashid, A.O. Akdemir, Y.M. Chu, New Hermite–Jensen–Mercer–Type Inequalities via K-Fractional Integrals, Adv. Differ. Equ. 2020 (2020), 635. https://doi.org/10.1186/s13662-020-03093-y.
  31. M. Tariq, S.K. Ntouyas, A.A. Shaikh, New Variant of Hermite–Hadamard, Fejér and Pachpatte-Type Inequality and Its Refinements Pertaining to Fractional Integral Operator, Fractal Fract. 7 (2023), 405. https://doi.org/10.3390/fractalfract7050405.
  32. R.L. Magin, Fractional Calculus in Bio-Engineering, Begell House Inc., 2006.
  33. A. Atangana, Application of Fractional Calculus to Epidemiology, in: C. Cattani, H.M. Srivastava, X.J. Yang (Eds.), Fractional Dynamics, De Gruyter Open Poland, Warsaw, Poland, 2015, pp. 174–190. https://doi.org/10.1515/9783110472097-011.
  34. D. Baleanu, A. Jajarmi, S.S. Sajjadi, D. Mozyrska, A New Fractional Model and Optimal Control of a Tumor-Immune Surveillance with Non-Singular Derivative Operator, Chaos 29 (2019), 083127. https://doi.org/10.1063/1.5096159.
  35. A. Ebrahimzadeh, A. Jajarmi, D. Baleanu, Enhancing Water Pollution Management Through a Comprehensive Fractional Modeling Framework and Optimal Control Techniques, J. Nonlinear Math. Phys. 31 (2024), 48. https://doi.org/10.1007/s44198-024-00215-y.
  36. D. Baleanu, A. Jajarmi, O. Defterli, R. Wannan, S.S. Sajjadi, et al., Fractional Investigation of Time-Dependent Mass Pendulum, J. Low Freq. Noise, Vib. Act. Control. 43 (2023), 196–207. https://doi.org/10.1177/14613484231187439.
  37. M. Axtell, M. Bise, Fractional Calculus Application in Control Systems, in: IEEE Conference on Aerospace and Electronics, IEEE, pp. 563–566. https://doi.org/10.1109/NAECON.1990.112826.
  38. } C.P. Niculescu, L.E. Persson, Convex Functions and Their Applications, Springer New York, 2006. https://doi.org/10.1007/0-387-31077-0.
  39. R.K. Raina, On Generalized Wright's Hypergeometric Functions and Fractional Calculus Operators, East Asian Math. J. 21 (2005), 191–203.
  40. M.J. Vivas-Cortez, R. Liko, A. Kashuri, J.E. Hernández Hernández, New Quantum Estimates of Trapezium-Type Inequalities for Generalized $phi$-Convex Functions, Mathematics 7 (2019), 1047. https://doi.org/10.3390/math7111047.
  41. M.J. Vivas-Cortez, A. Kashuri, J.E.H. Hernández, Trapezium-Type Inequalities for Raina’s Fractional Integrals Operator Using Generalized Convex Functions, Symmetry 12 (2020), 1034. https://doi.org/10.3390/sym12061034.
  42. H. Ahmad, M. Tariq, S.K. Sahoo, J. Baili, C. Cesarano, New Estimations of Hermite–Hadamard Type Integral Inequalities for Special Functions, Fractal Fract. 5 (2021), 144. https://doi.org/10.3390/fractalfract5040144.
  43. S. Rashid, İ. İşcan, D. Baleanu, Y.M. Chu, Generation of New Fractional Inequalities via n Polynomials s-Type Convexity with Applications, Adv. Differ. Equ. 2020 (2020), 264. https://doi.org/10.1186/s13662-020-02720-y.
  44. M.A. Noor, Hermite-Hadamard Integral Inequalities for Log-Preinvex Functions, J. Math. Anal. Approx. Theory 2 (2007), 126–131.
  45. I. Iscan, Construction of a New Class of Functions with Their Some Properties and Certain Inequalities: n-Fractional Polynomial Convex Functions, Miskolc Math. Notes 24 (2023), 1389. https://doi.org/10.18514/MMN.2023.4142.
  46. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
  47. S. Mubeen, G.M. Habibullah, $k$–Fractional Integrals and Application, Int. J. Contemp. Math. Sci. 7 (2012), 89–94.