TY  - JOUR
T1  - Hermite-Hadamard Inequalities Involving Fractional Conformable Integral Operators for the Product of Two Interval-Valued LR-Convex Functions
AU  - Kara, Hasan
AU  - Ustaoğlu, Ömer
AU  - Budak, Hüseyin
PY  - 2025
DA  - September
Y2  - 2025
DO  - 10.33434/cams.1665337
JF  - Communications in Advanced Mathematical Sciences
PB  - Emrah Evren KARA
WT  - DergiPark
SN  - 2651-4001
SP  - 136
EP  - 150
VL  - 8
IS  - 3
LA  - en
AB  - 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.
KW  - Fractional conformable integral
KW  - Hermite-Hadamard type inequalities
KW  - Interval-valued functions
KW  - LR-convexity
CR  - [1] S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
CR  - [2] R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, 1966.
CR  - [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
CR  - [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.
CR  - [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
CR  - [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
CR  - [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
CR  - [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
CR  - [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
CR  - [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
CR  - [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
CR  - [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.
CR  - [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
CR  - [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
CR  - [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
CR  - [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
CR  - [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
CR  - [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
CR  - [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.
CR  - [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
CR  - [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
CR  - [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.
CR  - [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
CR  - [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.
CR  - [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
CR  - [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
CR  - [27] R. E. Moore, R. B. Kearfott, M. J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, PA, 2009.
CR  - [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
CR  - [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
CR  - [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
CR  - [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
CR  - [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
CR  - [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
CR  - [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.
CR  - [35] B. G. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Coll., 6 (2003).
UR  - https://doi.org/10.33434/cams.1665337
L1  - https://dergipark.org.tr/en/download/article-file/4724361
ER  -