ENHANCED HERMITE–HADAMARD INEQUALITIES FOR INTERVAL–VALUED FUNCTIONS VIA GENERALIZED FRACTIONAL OPERATORS AND ARTIFICIAL NEURAL NETWORK PREDICTIONS

Rania Saadeh; Mohamed A. Barakat; Abd-Allah Hyder; Hüseyin Budak; Abdelraheem Mahmoud Aly; Mohamed Hafez

doi:10.11948/20260002

2026 Volume 16 Issue 5

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Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(5): 2720-2750

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Rania Saadeh, Mohamed A. Barakat, Abd-Allah Hyder, Hüseyin Budak, Abdelraheem Mahmoud Aly, Mohamed Hafez. ENHANCED HERMITE–HADAMARD INEQUALITIES FOR INTERVAL–VALUED FUNCTIONS VIA GENERALIZED FRACTIONAL OPERATORS AND ARTIFICIAL NEURAL NETWORK PREDICTIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2720-2750. doi: 10.11948/20260002

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Rania Saadeh, Mohamed A. Barakat, Abd-Allah Hyder, Hüseyin Budak, Abdelraheem Mahmoud Aly, Mohamed Hafez. ENHANCED HERMITE–HADAMARD INEQUALITIES FOR INTERVAL–VALUED FUNCTIONS VIA GENERALIZED FRACTIONAL OPERATORS AND ARTIFICIAL NEURAL NETWORK PREDICTIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2720-2750. doi: 10.11948/20260002

ENHANCED HERMITE–HADAMARD INEQUALITIES FOR INTERVAL–VALUED FUNCTIONS VIA GENERALIZED FRACTIONAL OPERATORS AND ARTIFICIAL NEURAL NETWORK PREDICTIONS

1.
Department of Applied Science, Ajlun College, Al-Balqa Applied University, Ajlun, Jordan
2.
Department of Basic Science, University College of Alwajh, University of Tabuk, Saudi Arabia
3.
Department of Mathematics, College of Science, King Khalid University, Box 9004, 61413 Abha, Saudi Arabia
4.
Department of Mathematics, Faculty of Arts and Sciences, Kocaeli University, Kocaeli 41001, Türkiye
5.
Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, Tamil Nadu, India
6.
Faculty of Engineering FEQS, INTI-IU-University, Nilai, Malaysia
7.
Faculty of Management, Shinawatra University, Pathum Thani, Thailand

Author Bio: Email: m.barakat@ut.edu.sa(M. A. Barakat); Email: abahahmed@kku.edu.sa(A.-A. Hyder); Email: hsyn.budak@gmail.com(H. Budak); Email: ababdallah@kku.edu.sa(A. M. Aly); Email: mohdahmed.hafez@newinti.edu.my(M. Hafez)
Corresponding author: Email: r.saadeh@bau.edu.jo(R. Saadeh)

Abstract

Abstract

In this study we explore the development of a branch of the theory of fractional integration by presenting a new methodological framework based on generalized fractional integrals, focusing on interval functions. This framework enables the more systematic and clear derivation of advanced Hermit-Hadamard inequalities suitable for convex functions. This work integrates Riemann-Liouville fractional inequalities with the classical form of Hermit-Hadamard inequalities for LR convex functions within a unified structure, eliminating the need for separate case studies or independent proofs for each inequality. This research also extends to the development of new Hermite-Hadamard inequalities targeting LR convex functions with interval values, further increasing the comprehensiveness of this proposed framework. The theoretical results obtained in this research are validated through discussions on examples and illustrations that show their application in the field of fractional integration inequalities. Artificial neural networks were also used in this research to correctly forecast the boundaries associated with inequalities in this field, increasing the reliability of this research's results.
- Hermite-Hadamard inequalities /
- fractional operators /
- ANNs predictions
MSC: 26D15, 26D10, 26D07

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References

[1]	A. El-Ajou, Developing the limit-residual function method for providing approximate solutions to initial and boundary-value problems, Fractals, 2026, 34(3), 2550127. doi: 10.1142/S0218348X25501270 CrossRef Google Scholar
[2]	I. Aldawish, M. Jleli and B. Samet, On hermite–hadamard-type inequalities for functions satisfying second-order differential inequalities, Axioms, 2023, 12(5), 443. doi: 10.3390/axioms12050443 CrossRef Google Scholar
[3]	M. A. Barakat, R. Saadeh, A. Hyder, et al., A novel fractional model combined with numerical simulation to examine the impact of lockdown on COVID-19 spread, Fractal and Fractional, 2024, 8(12), 702. doi: 10.3390/fractalfract8120702 CrossRef Google Scholar
[4]	W. W. Breckner, Continuity of generalized convex and generalized concave set-valued functions, Revue d'Analyse Numérique et de Théorie de l'Approximation, 1993, 22, 39–51. Google Scholar
[5]	H. Budak, T. Tunc and M. Z. Sarikaya, Fractional hermite-hadamard-type inequalities for interval–valued functions, Proceedings of the American Mathematical Society, 2020, 148, 705–718. Google Scholar
[6]	Y. Chalco-Cano, A. Flores-Franulič and H. Romún-Flores, Ostrowski type inequalities for interval–valued functions using generalized hukuhara derivative, Computational and Applied Mathematics, 2012, 31, 457–472. Google Scholar
[7]	Y. Chalco-Cano, W. A. Lodwick and W. Condori-Equice, Ostrowski type inequalities and applications in numerical integration for interval–valued functions, Soft Computing, 2015, 19, 3293–3300. doi: 10.1007/s00500-014-1483-6 CrossRef Google Scholar
[8]	F. Chen, A note on hermite–hadamard inequalities for products of convex functions, Journal of Applied Mathematics, 2013, 2013, Article ID 935020, 5 pp. Google Scholar
[9]	T. M. Costa, Jensen's inequality type integral for fuzzy-interval–valued functions, Fuzzy Sets and Systems, 2017, 327, 31–47. doi: 10.1016/j.fss.2017.02.001 CrossRef Google Scholar
[10]	T. M. Costa and H. Romún-Flores, Some integral inequalities for fuzzy-interval–valued functions, Information Sciences, 2017, 420, 110–125. doi: 10.1016/j.ins.2017.08.055 CrossRef Google Scholar
[11]	S. S. Dragomir, Some inequalities of Hermite–Hadamard type for symmetrized convex functions and Riemann–Liouville fractional integrals, RGMIA Research Report Collection, 2017, 20, 15. Google Scholar
[12]	S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. Google Scholar
[13]	S. S. Dragomir, J. Pecaric and L. E. Persson, Some inequalities of hadamard type, Soochow Journal of Mathematics, 1995, 21, 335–341. Google Scholar
[14]	T. Eriqat, M. N. Qiealat, Z. Al-Zhour, et al., Revisited fisher's equation and logistic system model: A new fractional approach and some modifications, International Journal of Dynamics and Control, 2023, 11, 555–563. doi: 10.1007/s40435-022-01020-5 CrossRef Google Scholar
[15]	H. Romún-Flores, Y. Chalco-Cano and W. A. Lodwick, Some integral inequalities for interval–valued functions, Computational and Applied Mathematics, 2018, 37, 1306–1318. doi: 10.1007/s40314-016-0396-7 CrossRef Google Scholar
[16]	H. Romún-Flores, Y. Chalco-Cano and G. N. Silva, A note on gronwall type inequality for interval–valued functions, in 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), IEEE, 2013, 1455–1458. Google Scholar
[17]	A. Flores-Franulič, Y. Chalco-Cano and H. Romún-Flores, An ostrowski type inequality for interval–valued functions, in 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), IEEE, 2013, 1459–1462. Google Scholar
[18]	R. Osuna-Gómez, M. D. Jiménez-Gamero, Y. Chalco-Cano and M. A. Rojas-Medar, Hadamard and jensen inequalities for $s$-convex fuzzy processes, in Soft Methodology and Random Information Systems, Springer, Berlin/Heidelberg, 2004, 645–652. Google Scholar
[19]	A. Gozpinar, Some hermite-hadamard type inequalities for convex functions via new fractional generalized integrals and related inequalities, in AIP Conf. Proc., 1991, 2018, 020006. Google Scholar
[20]	T. Hamadneh, A. Hioual, R. Saadeh, et al., General methods to synchronize fractional discrete reaction–diffusion systems applied to the glycolysis model, Fractal and Fractional, 2023, 7(11), 828. doi: 10.3390/fractalfract7110828 CrossRef Google Scholar
[21]	A. Hussain, M. Hammad, A. A. Rahimzai, et al., Dynamical analysis and soliton solutions of the space–time fractional kaup–boussinesq system, Partial Differential Equations in Applied Mathematics, 2025, 14, 101205. doi: 10.1016/j.padiff.2025.101205 CrossRef Google Scholar
[22]	A. Hyder, A. A. Almoneef, H. Budak and M. A. Barakat, On new fractional version of generalized hermite-hadamard inequalities, Mathematics, 2022, 10(18), 3337. doi: 10.3390/math10183337 CrossRef Google Scholar
[23]	F. Jarad, E. Uğurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Advances in Difference Equations, 2017, 2017, 247. doi: 10.1186/s13662-017-1306-z CrossRef Google Scholar
[24]	I. Khan, A. Ahmad, H. Khan and T. Abdeljawad, Existence and stability analysis for a nabla discrete abc fractional COVID-19 model, Advances in Difference Equations, 2021, 2021(1), 1–17. doi: 10.1186/s13662-020-03162-2 CrossRef Google Scholar
[25]	M. B. Khan, P. O. Mohammed, M. A. Noor and Y. S. Hamed, New hermite-hadamard inequalities in fuzzy-interval fractional calculus and related inequalities, Symmetry, 2021, 13, 673. doi: 10.3390/sym13040673 CrossRef Google Scholar
[26]	M. B. Khan, P. O. Mohammed, M. A. Noor, et al., Some new fractional estimates of inequalities for $lr$-$p$-convex interval–valued functions by means of pseudo order relation, Axioms, 2021, 10, 175. doi: 10.3390/axioms10030175 CrossRef $lr$-$p$-convex interval–valued functions by means of pseudo order relation" target="_blank">Google Scholar
[27]	M. B. Khan, M. A. Noor, T. Abdeljawad, et al., $lr$-preinvex interval–valued functions and r-l fractional integral inequalities, Fractal and Fractional, 2021, 5, 243. doi: 10.3390/fractalfract5040243 CrossRef $lr$-preinvex interval–valued functions and r-l fractional integral inequalities" target="_blank">Google Scholar
[28]	M. B. Khan, H. M. Srivastava, P. O. Mohammed, et al., Fuzzy-interval inequalities for generalized preinvex fuzzy interval valued functions, Mathematical Biosciences and Engineering, 2022, 19, 812–835. Google Scholar
[29]	M. B. Khan, S. Treant, M. S. Soliman, et al., Some Hadamard-Fejér type inequalities for $lr$-convex interval–valued functions, Fractal and Fractional, 2022, 6(6). $lr$-convex interval–valued functions" target="_blank">Google Scholar
[30]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, 204 of North-Holland Mathematics Studies, Elsevier Sci. B.V., Amsterdam, 2006. Google Scholar
[31]	D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, in International Conference on Learning Representations (ICLR), 2014. Google Scholar
[32]	S. W. Lee, M. Alotaibi and A. M. Aly, Integrating isph simulations and artificial neural networks for simulating free surface flow over various porous media on slopes, Computational Particle Mechanics, 2023. Google Scholar
[33]	K. Levenberg, A method for the solution of certain nonlinear problems in least squares, Quarterly of Applied Mathematics, 1944, 2, 164–168. doi: 10.1090/qam/10666 CrossRef Google Scholar
[34]	X. Liu, G. Ye, D. Zhao and W. Liu, Fractional hermite-hadamard type inequalities for interval–valued functions, Journal of Inequalities and Applications, 2019, 2019, 1–11. Google Scholar
[35]	V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets and Systems, 2015, 265, 63–85. doi: 10.1016/j.fss.2014.04.005 CrossRef Google Scholar
[36]	D. W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics, 1963, 11, 431–441. doi: 10.1137/0111030 CrossRef Google Scholar
[37]	F. -C. Mitroi, K. Nikodem and S. Wasowicz, Hermite–hadamard inequalities for convex set-valued functions, Demonstratio Mathematica, 2013, 46, 655–662. Google Scholar
[38]	R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, 1966. Google Scholar
[39]	R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to interval analysis, SIAM, Philadelphia, P.A., 2009. Google Scholar
[40]	K. Nikodem, J. L. Sanchez and L. Sanchez, Jensen and hermite-hadamard inequalities for strongly convex set-valued maps, Mathematica Aeterna, 2014, 4, 979–987. Google Scholar
[41]	M. N. Oqielat, T. Eriqat, Z. Al-Zhour, et al., Numerical solutions of time-fractional nonlinear water wave partial differential equation via caputo fractional derivative: An effective analytical method and some applications, Applied and Computational Mathematics, 2022, 21(2), 207–222. Google Scholar
[42]	B. G. Pachpatte, On some inequalities for convex functions, RGMIA Research Report Collection, 2003, 6. Google Scholar
[43]	Z. Pavic, Improvements of the hermite–hadamard inequality, Journal of Inequalities and Applications, 2015, 2015, 222. doi: 10.1186/s13660-015-0742-0 CrossRef Google Scholar
[44]	J. E. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992. Google Scholar
[45]	J. Periasamy, P. Suthanthira Devi and D. A. Dew, Optimized deep learning for car insurance fraud detection using gray wolf algorithm, in Proceedings of the 2025 International Conference on Data Science and Business Systems (ICDSBS), IEEE, 2025, 1–6. Google Scholar
[46]	E. Sadowska, Hadamard inequality and a refinement of jensen inequality for set valued functions, Results Math., 1997, 32, 332–337. doi: 10.1007/BF03322144 CrossRef Google Scholar
[47]	E. Salah, A. Qazza, R. Saadeh and A. El-Ajou, A hybrid analytical technique for solving multi-dimensional time-fractional navier–stokes system, AIMS Mathematics, 2022, 8(1), 1713–1736. Google Scholar
[48]	B. Samet, A convexity concept with respect to a pair of functions, Numerical Functional Analysis and Optimization, 2022, 43(5), 522–540. doi: 10.1080/01630563.2022.2050753 CrossRef Google Scholar
[49]	G. Sana, M. B. Khan, M. A. Noor, et al., Harmonically convex fuzzy-interval–valued functions and fuzzy-interval riemann-liouville fractional integral inequalities, International Journal of Computational Intelligence Systems, 2021, 14, 1809–1822. doi: 10.2991/ijcis.d.210620.001 CrossRef Google Scholar
[50]	M. Z. Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite–hadamard's inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 2013, 57(9–10), 2403–2407. Google Scholar
[51]	M. Z. Sarikaya and H. Yildirim, On hermite-hadamard type inequalities for riemann-liouville fractional integrals, Miskolc Mathematical Notes, 2016, 17(2), 1049–1059. Google Scholar
[52]	E. Set, J. Choi and A. G. Nar, Hermite–hadamard type inequalities for new generalized fractional integral operator, Malaysian Journal of Mathematical Sciences, 2021, 15, 33–43. Google Scholar
[53]	K. L. Tseng and S. R. Hwang, New hermite–hadamard inequalities and their applications, Filomat, 2016, 30(14), 3667–3680. doi: 10.2298/FIL1614667T CrossRef Google Scholar
[54]	M. S. Zahoor, A. Hussain and Y. Wang, New fractional Hermite–Hadamard-type inequalities for Caputo derivative and MET-$(p, s)$-convex functions with applications, Fractal and Fractional, 2026, 10(1), 62. doi: 10.3390/fractalfract10010062 CrossRef $(p, s)$-convex functions with applications" target="_blank">Google Scholar
[55]	D. Zhang, G. C. Guo, D. Chen and G. Wang, Jensen's inequalities for set-valued and fuzzy set-valued functions, Fuzzy Sets and Systems, 2021, 404, 178–204. doi: 10.1016/j.fss.2020.06.003 CrossRef Google Scholar
[56]	L. Zhang, M. Feng, R. P. Agarwal and G. Wang, Concept and application of interval-valued fractional generalized calculus, Alexandria Engineering Journal, 2022, 61(12), 11959–11977. doi: 10.1016/j.aej.2022.06.005 CrossRef Google Scholar
[57]	D. Zhao, T. An, G. Ye and W. Liu, New jensen and hermite-hadamard type inequalities for $h$-convex interval–valued functions, Journal of Inequalities and Applications, 2018, 2018, 302. doi: 10.1186/s13660-018-1896-3 CrossRef $h$-convex interval–valued functions" target="_blank">Google Scholar
[58]	D. Zhao, T. An, G. Ye and W. Liu, Chebyshev type inequalities for interval–valued functions, Fuzzy Sets and Systems, 2020, 396, 82–101. doi: 10.1016/j.fss.2019.10.006 CrossRef Google Scholar
[59]	D. Zhao, G. Ye, W. Liu and D. F. M. Torres, Some inequalities for interval–valued functions on time scales, Soft Computing, 2019, 23, 6005–6015. Google Scholar