EXPLORING LOCAL FRACTIONAL INTEGRAL INEQUALITIES THROUGH BIPARAMETRIC ANALYSIS

Thabet Abdeljawad; Abdelghani Lakhdari; Manar. A. Alqudah; Hisham M. Alkhawar; Badreddine Meftah

doi:10.11948/20260138

2027 Volume 17 Issue 2

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Thabet Abdeljawad, Abdelghani Lakhdari, Manar. A. Alqudah, Hisham M. Alkhawar, Badreddine Meftah. EXPLORING LOCAL FRACTIONAL INTEGRAL INEQUALITIES THROUGH BIPARAMETRIC ANALYSIS[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1649-1676. doi: 10.11948/20260138

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Thabet Abdeljawad, Abdelghani Lakhdari, Manar. A. Alqudah, Hisham M. Alkhawar, Badreddine Meftah. EXPLORING LOCAL FRACTIONAL INTEGRAL INEQUALITIES THROUGH BIPARAMETRIC ANALYSIS[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1649-1676. doi: 10.11948/20260138

EXPLORING LOCAL FRACTIONAL INTEGRAL INEQUALITIES THROUGH BIPARAMETRIC ANALYSIS

1.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
2.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa
3.
Department of Fundamental Sciences, Faculty of Engineering and Architecture, Istanbul Gelisim University, Avcılar-Istanbul 34310, Türkiye
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
5.
Department of Computer Science and Information Engineering, Asia University, Taichung 41354, Taiwan
6.
National Higher School of Technology and Engineering, Laboratory of Energy Systems Technologies (LTSE) E3360100, Annaba 23005, Algeria
7.
Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, India
8.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84424, Riyadh 11671, Saudi Arabia
9.
Preparatory Year Program, Computer Department, Prince Sultan University, 11586 Riyadh, Saudi Arabia
10.
Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, Guelma 24000, Algeria

Author Bio: Email: tabdeljawad@gelisim.edu.tr(T. Abdeljawad); Email: maalqudah@pnu.edu.sa(M. A. Alqudah); Email: halkhawar@psu.edu.sa(H. M. Alkhawar); Email: badrimeftah@yahoo.fr(B. Meftah)
Corresponding author: Email: a.lakhdari@ensti-annaba.dz(A. Lakhdari)

Abstract

Abstract

In this paper, we investigate a class of biparametrized integral inequalities in the setting of local fractional calculus on fractal sets. Motivated by earlier results in [27, 39], we derive a new integral identity associated with a three-point Newton–Cotes formula involving two parameters. A noteworthy feature of this formula is that, through suitable choices of the parameters, it unifies a broad family of one-point, two-point, and three-point quadrature rules. Based on this identity, we establish several new inequalities for functions whose first-order local fractional derivatives satisfy generalized s-convexity assumptions. Our results not only extend a number of existing inequalities in the literature, but also recover several known estimates as particular cases. To illustrate the effectiveness of the theoretical findings, we present an example supported by two- and three-dimensional graphical representations, together with some applications related to quadrature formulas and special means.
- Newton–Cotes formula /
- biparametrized identity /
- local fractional integral /
- generalized s-convex functions /
- fractal set
MSC: 26D10, 26A51, 26D15

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References

[1]	T. Abdeljawad, S. Rashid, Z. Hammouch and Y. -M. Chu, Some new local fractional inequalities associated with generalized (s, m)convex functions and applications, Adv. Difference Equ. 2020, Paper No. 406, 27 pp. https://doi.org/10.1186/s13662-020-02865-w. doi: 10.1186/s13662-020-02865-w CrossRef Google Scholar
[2]	A. J. Acevedo and Juan E. Nápoles Valdés, On the Hermite-Hadamard inequality. Some methodological remarks, Phys Astron Int J., 2025, 9(2), 80–86. https://doi.org/10.15406/paij.2025.09.00368. doi: 10.15406/paij.2025.09.00368 CrossRef Google Scholar
[3]	S. I. Butt, D. Khan, P. Agarwal and B. Bin-Mohsin, Fractal-fractional perspectives of trapezoid and Bullen type bounds with applications, Mathematical Methods in the Applied Sciences, 2026, 1–33, https://doi.org/10.1002/mma.70555. doi: 10.1002/mma.70555 CrossRef Google Scholar
[4]	S. I. Butt, D. Khan and Y. Seol, Fractal perspective of superquadratic functions with generalized probability estimations, PloS one 20.2, 2025, e0313361. https://doi.org/10.1371/journal.pone.0313361. doi: 10.1371/journal.pone.0313361 CrossRef Google Scholar
[5]	S. I. Butt, D. Khan, S. Tipurić-Spužević and Y. Seol, Generalized Petrovič inequality and its related estimations in superquadraticity on the fractal space with applications, Fractals, 2025, 33(09), 2550086. https://doi.org/10.1142/S0218348X25500860 doi: 10.1142/S0218348X25500860 CrossRef Google Scholar
[6]	G. -S. Chen, Generalizations of Hölder's and some related integral inequalities on fractal space, J. Funct. Spaces Appl., 2013, 198405, 9 pp. http://dx.doi.org/10.1155/2013/198405. Google Scholar
[7]	T. Du, H. Wang, M. A. Khan and Y. Zhang, Certain integral inequalities considering generalized m-convexity on fractal sets and their applications, Fractals, 2019, 27(7), 1950117, 17 pp. https://doi.org/10.1142/S0218348X19501172. doi: 10.1142/S0218348X19501172 CrossRef Google Scholar
[8]	T. Du and X. Yuan, On the parameterized fractal integral inequalities and related applications, Chaos, Solitons & Fractals, 2023, 170, 113375. https://doi.org/10.1016/j.chaos.2023.113375. doi: 10.1016/j.chaos.2023.113375 CrossRef Google Scholar
[9]	T. S. Du and Y. Peng, Hermite–Hadamard type inequalities for multiplicative Riemann– Liouville fractional integrals, J. Comput. Appl. Math., 2024, 440, 115582. https://doi.org/10.1016/j.cam.2023.115582. doi: 10.1016/j.cam.2023.115582 CrossRef Google Scholar
[10]	M. Eddahbi, A. Lakhdari, B. Meftah, L. Mchiri and M. Rhaima, A comprehensive analysis of Riemann-Liouville fractional multiplicative integral inequalities, AIMS Math., 2025, 10, 11, 25227–25252. https://www.aimspress.com/article/doi/10.3934/math.20251117. doi: 10.3934/math.20251117 CrossRef Google Scholar
[11]	S. Erden and M. Z. Sarikaya, Generalized Bullen type inequalities for local fractional integrals and its applications, Palest. J. Math., 2020, 9(2), 945–956. Google Scholar
[12]	K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, 2004. Google Scholar
[13]	F. Gao, W. P. Zhong and X. M. Shen, Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral, Adv. Mater. Res., 2012, 461, 306–310. https://doi.org/10.4028/www.scientific.net/AMR.461.306. doi: 10.4028/www.scientific.net/AMR.461.306 CrossRef Google Scholar
[14]	Z. A. Khan, S. Rashid, R. Ashraf, D. Baleanu and Y. -M. Chu, Generalized trapezium-type inequalities in the settings of fractal sets for functions having generalized convexity property, Adv. Difference Equ., 2020, Paper No. 657, 24 pp. https://doi.org/10.1186/s13662-020-03121-x. doi: 10.1186/s13662-020-03121-x CrossRef Google Scholar
[15]	A. Lakhdari, W. Saleh, B. Meftah and A. Iqbal, Corrected dual Simpson type inequalities for differentiable generalized convex functions on fractal set, Fractal Fract., 2022, 6(12), 710. https://doi.org/10.3390/fractalfract6120710. doi: 10.3390/fractalfract6120710 CrossRef Google Scholar
[16]	C. Luo, H. Wang and T. Du, Fejér-Hermite–Hadamard type inequalities involving generalized h-convexity on fractal sets and their applications, Chaos Solitons Fractals, 2020, 131, 109547, 13 pp. Google Scholar
[17]	B. B. Mandelbrot, The Fractal Geometry of Nature, Vol. 1, New York, WH Freeman, 1982. Google Scholar
[18]	B. Meftah, D. C. Benchettah and A. Lakhdari, Some new local fractional Newton-type inequalities, Journal of Inequalities and Mathematical Analysis, 2025, 1(2), 79–96. https://doi.org/10.63286/jima.008. doi: 10.63286/jima.008 CrossRef Google Scholar
[19]	B. Meftah, A. Lakhdari and D. C. Benchettah, Some new Hermite-Hadamard type integral inequalities for twice differentiable s-convex functions, Comput. Math. Model., 2022, 33(3), 330–353. https://doi.org/10.1007/s10598-023-09576-3. doi: 10.1007/s10598-023-09576-3 CrossRef Google Scholar
[20]	B. Meftah, A. Lakhdari, W. Saleh and A. Kiliçman, Some new fractal Milne type integral inequalities via generalized convexity with applications, Fractal Fract., 2023, 7(2), 166. https://doi.org/10.3390/fractalfract7020166. doi: 10.3390/fractalfract7020166 CrossRef Google Scholar
[21]	B. Meftah, A. Souahi and M. Merad, Some local fractional Maclaurin type inequalities for generalized convex functions and their applications, Chaos Solitons Fractals, 2022, 162, Paper No. 112504, 7 pp. Google Scholar
[22]	H. Mo and X. Sui, Hermite–Hadamard-type inequalities for generalized s-convex functions on real linear fractal set $\mathbb{R}.{\alpha} $ $(0<\alpha<1) $, Math. Sci., 2017, 11, 241–246. https://doi.org/10.1007/s40096-017-0227-z. doi: 10.1007/s40096-017-0227-z CrossRef $\mathbb{R}.{\alpha} $ $(0<\alpha<1) $" target="_blank">Google Scholar
[23]	H. -X. Mo and X. Sui, Generalized s-convex functions on fractal sets, Abstr. Appl. Anal., 2014, 8, 1–3. https://doi.org/10.1155/2014/254737. doi: 10.1155/2014/254737 CrossRef Google Scholar
[24]	J. E. Nápoles Valdés, A review of Hermite-Hadamard inequality, Partners Universal International Rese Arch Journal (PUIRJ), 2022, 1(4), 98–101. https://doi.org/10.5281/zenodo.7492608. doi: 10.5281/zenodo.7492608 CrossRef Google Scholar
[25]	J. E. Nápoles Valdés, F. Rabossi and A. D. Samaniego, Convex functions: Ariadne's thread or Charlotte spiderweb? Advanced Mathematical Models & Applications, 2020, 5(2), 176–191. Google Scholar
[26]	J. E. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering, 187, Academic Press, Inc., Boston, MA, 1992. Google Scholar
[27]	W. Saleh, A. Lakhdari, T. Abdeljawad and B. Meftah, On fractional biparameterized Newton-type inequalities, J. Inequal. Appl., 2023, Paper No. 122, 18 pp. https://doi.org/10.1186/s13660-023-03033-w. doi: 10.1186/s13660-023-03033-w CrossRef Google Scholar
[28]	W. Saleh, A. Lakhdari, O. Almutairi and A. Kiliçman, Some remarks on local fractional integral inequalities involving Mittag–Leffler kernel using generalized $(E, h) $-convexity, Mathematics, 2023, 11(6), 1373. https://doi.org/10.3390/math11061373. doi: 10.3390/math11061373 CrossRef $(E, h) $-convexity" target="_blank">Google Scholar
[29]	W. Saleh, B. Meftah, A. Lakhdari and A. Kiliçman, Exploring the companion of Ostrowski's inequalities via local fractional integrals, European Journal of Pure and Applied Mathematics, 2023, 16(3), 1359–1380. https://doi.org/10.29020/nybg.ejpam.v16i3.4850. doi: 10.29020/nybg.ejpam.v16i3.4850 CrossRef Google Scholar
[30]	M. Z. Sarikaya, H. Budak and S. Erden, On new inequalities of Simpson's type for generalized convex functions, Korean J. Math., 2019, 27(2), 279–295. https://doi.org/10.11568/kjm.2019.27.2.279. doi: 10.11568/kjm.2019.27.2.279 CrossRef Google Scholar
[31]	E. Set and M. Tomar, New inequalities of Hermite–Hadamard type for generalized convex functions with applications, Facta Univ. Ser. Math. Inform., 2016, 31(2), 383–397. Google Scholar
[32]	J. Shang, Y. Wang, M. Chen, J. Dai, X. Zhou, J. Kuttner, G. Hilt, X. Shao, J. M. Gottfried and K. Wu, Assembling molecular Sierpiński triangle fractals, Nature Chemistry, 2015, 7(5), 389–393. https://doi.org/10.1038/nchem.2211. doi: 10.1038/nchem.2211 CrossRef Google Scholar
[33]	W. Sun, Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, Fractals, 2019, 27(5), 1950071, 14 pp. https://doi.org/10.1142/S0218348X19500713. doi: 10.1142/S0218348X19500713 CrossRef Google Scholar
[34]	H. Xu, A. Lakhdari, W. Saleh and B. Meftah, Some new parametrized inequalities on fractal set, Fractals, 2024, 32(03), 2450063. https://doi.org/10.1142/S0218348X24500634. doi: 10.1142/S0218348X24500634 CrossRef Google Scholar
[35]	A. M. Yang, Z. S. Chen, H. M. Srivastava and X. J. Yang, Application of the local fractional series expansion method and the variational iteration method to the Helmholtz equation involving local fractional derivative operators, Abstr. Appl. Anal., (2013), Article ID 259125. https://doi.org/10.1155/2013/259125. doi: 10.1155/2013/259125 CrossRef Google Scholar
[36]	X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, 2012. Google Scholar
[37]	Y. J. Yang, D. Baleanu and X.-J. Yang, Analysis of fractal wave equations by local fractional Fourier series method, Adv. Math. Phys., 2013, Art. ID 632309, 6 pp. https://doi.org/10.1155/2013/632309. doi: 10.1155/2013/632309 CrossRef Google Scholar
[38]	Y. Yu and T. Du, Certain error bounds on the Bullen type integral inequalities in the framework of fractal spaces, J. Nonlinear Funct. Anal., 2022, 2022, 24. https://doi.org/10.23952/jnfa.2022.24. doi: 10.23952/jnfa.2022.24 CrossRef Google Scholar
[39]	Y. Yu, J. Liu and T. Du, Certain error bounds on the parameterized integral inequalities in the sense of fractal sets, Chaos, Solitons & Fractals, 2022, 161, 112328. https://doi.org/10.1016/j.chaos.2022.112328. doi: 10.1016/j.chaos.2022.112328 CrossRef Google Scholar
[40]	L. L. Zhang, Y. Peng and T. S. Du, On multiplicative Hermite–Hadamard- and Newtontype inequalities for multiplicatively (P, m)-convex functions, J. Math. Anal. Appl., 2024, 534(2), 128117. https://doi.org/10.1016/j.jmaa.2024.128117. doi: 10.1016/j.jmaa.2024.128117 CrossRef Google Scholar
[41]	Y. X. Zhou and T. S. Du, The Simpson-type integral inequalities involving twice local fractional differentiable generalized $(s, P) $-convexity and their applications, Fractals, 2023, 31(5), 2350038. https://doi.org/10.1142/S0218348X2350038X. doi: 10.1142/S0218348X2350038X CrossRef $(s, P) $-convexity and their applications" target="_blank">Google Scholar