Hermite-Hadamard Inequalities Involving Fractional Conformable Integral Operators for the Product of Two Interval-Valued LR-Convex Functions

Year 2025, Volume: 8 Issue: 3 , 136 - 150 , 23.09.2025

Ömer Ustaoğlu , Hasan Kara , Hüseyin Budak

https://doi.org/10.33434/cams.1665337

https://izlik.org/JA75HD75HY

Abstract

In this research, new Hermite-Hadamard-type inequalities are obtained for the product of two LR-convex intervalvalued functions utilizing fractional conformable integrals. Specific parameter choices are employed to generalize existing results and to acquire new findings in the field. Examples are provided to illustrate the truth of the established inequalities, and graphical representations are included to support the understanding of these examples further.

Keywords

Fractional conformable integral , Hermite-Hadamard type inequalities , Interval-valued functions , LR-convexity

References

• [1] S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
• [2] R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, 1966.
• [3] E. Sadowska, Hadamard inequality and a refinement of Jensen inequality for set valued functions, Results Math., 32(3-4) (1997), 332–337. https://doi.org/10.1007/BF03322144
• [4] Y. Chalco-Cano, A. Flores-Franuliˇc, H. Rom´an-Flores, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457–472.
• [5] A. Flores-Franuliˇc, Y. Chalco-Cano, H. Rom´an-Flores, An Ostrowski type inequality for interval-valued functions, In: Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), IEEE, 2013, pp. 1459–1462. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608617
• [6] H. Rom´an-Flores, Y. Chalco-Cano, W. A. Lodwick, Some integral inequalities for interval-valued functions, Comput. Appl. Math., 37(2) (2018), 1306–1318. https://doi.org/10.1007/s40314-016-0396-7
• [7] H. Budak, T. Tunc, M. Z. Sarikaya, Fractional Hermite-Hadamard-type inequalities for interval-valued functions, Proc. Amer. Math. Soc., 148 (2020), 705–718. https://doi.org/10.1090/proc/14741
• [8] X. Liu, G. Ye, D. Zhao, W. Liu, Fractional Hermite-Hadamard type inequalities for interval-valued functions, J. Inequal. Appl., 2019 (2019), Article ID 266. https://doi.org/10.1186/s13660-019-2217-1
• [9] T. M. Costa, Jensen’s inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets Syst., 327 (2017), 31–47. https://doi.org/10.1016/j.fss.2017.02.001
• [10] T. M. Costa, H. Rom´an-Flores, Some integral inequalities for fuzzy-interval-valued functions, Inform. Sci., 420 (2017), 110–125. https://doi.org/10.1016/j.ins.2017.08.055
• [11] F. C. Mitroi, K. Nikodem, S. Wasowicz, Hermite–Hadamard inequalities for convex set-valued functions, Demonstr. Math., 46(4) (2013), 655–662. https://doi.org/10.1515/dema-2013-0483
• [12] K. Nikodem, J. L. Sanchez, L. Sanchez, Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps, Math. AEterna, 4(8) (2014), 979–987.
• [13] H. Roman-Flores, Y. Chalco-Cano, G. N. Silva, A note on Gronwall type inequality for interval-valued functions, In: Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), IEEE, 2013, pp. 1455–1458. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608616
• [14] D. Zhao, T. An, G. Ye, W. Liu, Chebyshev type inequalities for interval-valued functions, Fuzzy Sets Syst., 396 (2020), 82–101. https://doi.org/10.1016/j.fss.2019.10.006
• [15] D. Zhao, G. Ye, W. Liu, D. F. M. Torres, Some inequalities for interval-valued functions on time scales, Soft Comput., 23(15) (2019), 6005–6015. https://doi.org/10.1007/s00500-018-3538-6
• [16] M. B. Khan, P. O. Mohammed, M. A. Noor, Y. S. Hamed, New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus and related inequalities, Symmetry, 13(4) (2021), Article ID 673. https://doi.org/10.3390/sym13040673
• [17] M. B. Khan, M. A. Noor, T. Abdeljawad, A. A. A. Mousa, B. Abdalla, S. M. Alghamdi, LR-preinvex intervalvalued functions and Riemann-Liouville fractional integral inequalities, Fractal Fract., 5(4) (2021), Article ID 243. https://doi.org/10.3390/fractalfract5040243
• [18] G. Sana, M. B. Khan, M. A. Noor, P. O. Mohammed, Y. M. Chu, Harmonically convex fuzzy-interval-valued functions and fuzzy-interval Riemann–Liouville fractional integral inequalities, Int. J. Comput. Intell. Syst., 14(1) (2021), 1809–1822. https://doi.org/10.2991/ijcis.d.210620.001
• [19] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.
• [20] M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57(9–10) (2013), 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048
• [21] M. Z. Sarikaya, H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17(2) (2016), 1049–1059. https://doi.org/10.18514/MMN.2017.1197
• [22] S. S. Dragomir, Some inequalities of Hermite-Hadamard type for symmetrized convex functions and Riemann-Liouville fractional integrals, RGMIA Res. Rep. Coll., 20 (2017), Article ID 46.
• [23] F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), Article ID 247. https://doi.org/10.1186/s13662-017-1306-z
• [24] E. Set, J. Choi, A. Gözpınar, Hermite–Hadamard type inequalities for new conformable fractional integral operator, Malays. J. Math. Sci., 15 (2021), 33–43.
• [25] A. Gozpinar, Some Hermite-Hadamard type inequalities for convex functions via new fractional conformable integrals and related inequalities, AIP Conf. Proc., 1991 (2018), Article ID 020006. https://doi.org/10.1063/1.5047879
• [26] A. Hyder, A. A. Almoneef, H. Budak, M. A. Barakat, On new fractional version of generalized Hermite-Hadamard inequalities, Mathematics, 10(18) (2022), Article ID 3337. https://doi.org/10.3390/math10183337
• [27] R. E. Moore, R. B. Kearfott, M. J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, PA, 2009.
• [28] D. Zhao, T. An, G. Ye,W. Liu, New Jensen and Hermite–Hadamard type inequalities for h-convex interval-valued functions, J. Inequal. Appl., 2018 (2018), Article ID 302. https://doi.org/10.1186/s13660-018-1896-3
• [29] V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets Syst., 265 (2015), 63–85. https://doi.org/10.1016/j.fss.2014.04.005
• [30] D. Zhang, G. C. Guo, D. Chen, G. Wang, Jensen’s inequalities for set-valued and fuzzy set-valued functions, Fuzzy Sets Syst., 404 (2021), 178–204. https://doi.org/10.1016/j.fss.2020.06.003
• [31] M. B. Khan, S. Treant, M. S. Soliman, K. Nonlaopon, H. G. Zaini, Some Hadamard-Fej´er type inequalities for LR-convex interval-valued functions, Fractal Fract., 6(1) (2022), Article ID 6. https://doi.org/10.3390/fractalfract6010006
• [32] H. M. Srivastava, S. K. Sahoo, P. O. Mohammed, B. Kodamasingh, Y. S. Hamed, New Riemann–Liouville fractional-order inclusions for convex functions via interval-valued settings associated with pseudo-order relations, Fractal Fract., 6(4) (2022), Article ID 212. https://doi.org/10.3390/fractalfract6040212
• [33] L. Zhang, M. Feng, R. P. Agarwal, G. Wang, Concept and application of interval-valued fractional conformable calculus, Alex. Eng. J., 61(12) (2022), 11959–11977. https://doi.org/10.1016/j.aej.2022.06.005
• [34] H. Kara, H. Budak, F. Hezenci, Hermite–Hadamard type inequalities for LR-convex interval-valued functions via fractional conformable integrals, In: Proceedings of the 9th International Conference on Control and Optimization with Industrial Applications (COIA 2024), Istanbul University-Cerrahpas¸a, 2024.
• [35] B. G. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Coll., 6 (2003).