Generalized fractional Hermite-Hadamard-type inequalities for interval-valued s-convex functions

Guoshan Deng; Dafang Zhao; Sina Etemad; Jessada Tariboon; Guoshan Deng; Dafang Zhao; Sina Etemad; Jessada Tariboon

doi:10.3934/math.2025635

AIMS Mathematics

2025, Volume 10, Issue 6: 14102-14121. doi: 10.3934/math.2025635

Previous Article Next Article

Research article Special Issues

Generalized fractional Hermite-Hadamard-type inequalities for interval-valued s-convex functions

1.
Huangshi Key Laboratory of Metaverse and Virtual Simulation, School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, China
2.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
3.
Mathematics in Applied Sciences and Engineering Research Group, Scientific Research Center, Al-Ayen University, Nasiriyah 64001, Iraq
4.
Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand

Received: 28 February 2025 Revised: 06 June 2025 Accepted: 10 June 2025 Published: 19 June 2025
MSC : 26E70, 34N05, 35A23

Based on the generalized fractional integrals, we develop some Hermite-Hadamard-type inequalities for interval-valued s-convex functions. Further, we verify our results with graphical representation and concrete examples. Our research not only generalizes and extends the existing literature but also provides valuable insights for further research on interval-valued integral inequalities.
- fractional integral,
- interval-valued function,
- Hermite-Hadamard inequalities,
- s-convex function
Citation: Guoshan Deng, Dafang Zhao, Sina Etemad, Jessada Tariboon. Generalized fractional Hermite-Hadamard-type inequalities for interval-valued s-convex functions[J]. AIMS Mathematics, 2025, 10(6): 14102-14121. doi: 10.3934/math.2025635

Related Papers:

Abstract

Based on the generalized fractional integrals, we develop some Hermite-Hadamard-type inequalities for interval-valued s-convex functions. Further, we verify our results with graphical representation and concrete examples. Our research not only generalizes and extends the existing literature but also provides valuable insights for further research on interval-valued integral inequalities.

References

[1]	T. X. Li, D. Acosta-Soba, A. Columbu, G. Viglialoro, Dissipative gradient nonlinearities prevent $\delta$-formations in local and nonlocal attraction-repulsion chemotaxis models, Stud. Appl. Math., 154 (2025), e70018. https://doi.org/10.1111/sapm.70018 doi: 10.1111/sapm.70018
[2]	T. X. Li, S. Frassu, G. Viglialoro, Combining effects ensuring boundedness in an attraction- repulsion chemotaxis model with production and consumption, Z. Angew. Math. Phys., 74 (2023), 109. https://doi.org/10.1007/s00033-023-01976-0 doi: 10.1007/s00033-023-01976-0
[3]	S. S. Dragomir, New inequalities of Hermite-Hadamard type for log-convex functions, Khayyam J. Math., 3 (2017), 98–115. https://doi.org/10.22034/kjm.2017.47458 doi: 10.22034/kjm.2017.47458
[4]	C. P. Niculescu, The Hermite-Hadamard inequality for log-convex functions, Nonlinear Anal., 75 (2012), 662–669. https://doi.org/10.1016/j.na.2011.08.066 doi: 10.1016/j.na.2011.08.066
[5]	I. Iscan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935–942.
[6]	I. Iscan, S. H. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238 (2014), 237–244. https://doi.org/10.1016/j.amc.2014.04.020 doi: 10.1016/j.amc.2014.04.020
[7]	M. Z. Sarikaya, A. Saglam, H. Yildirim, On some Hadamard-type inequalities for $h$-convex functions, J. Math. Inequal., 2 (2008), 335–341. https://doi.org/10.7153/jmi-02-30 doi: 10.7153/jmi-02-30
[8]	S. Varosanec, On $h$-convexity, J. Math. Anal. Appl., 326 (2007), 303–311. https://doi.org/10.1016/j.jmaa.2006.02.086
[9]	D. F. Zhao, T. Q. An, G. J. Ye, W. Liu, New Jensen and Hermite-Hadamard type inequalities for $h$-convex interval-valued functions, J. Inequal. Appl., 2018 (2018), 1–14. https://doi.org/10.1186/s13660-018-1896-3 doi: 10.1186/s13660-018-1896-3
[10]	S. I. Butt, M. N. Aftab, H. A. Nabwey, S. Etemad, Some Hermite-Hadamard and midpoint type inequalities in symmetric quantum calculus, AIMS Math., 9 (2024), 5523–5549. http://doi.org/10.3934/math.2024268 doi: 10.3934/math.2024268
[11]	T. Sitthiwirattham, M. A. Ali, H. Budak, S. Etemad, S. Rezapour, A new version of $(p, q)$-Hermite-Hadamard's midpoint and trapezoidal inequalities via special operators in $(p, q)$-calculus, Bound. Value Probl., 2022 (2022), 84. https://doi.org/10.1186/s13661-022-01665-3 doi: 10.1186/s13661-022-01665-3
[12]	S. Chasreechai, M. A. Ali, M. A. Ashraf, T. Sitthiwirattham, S. Etemad, M. De la Sen, et al., On new estimates of $q$-Hermite-Hadamard inequalities with applications in quantum calculus, Axioms, 12 (2023), 1–14. https://doi.org/10.3390/axioms12010049 doi: 10.3390/axioms12010049
[13]	H. Hudzik, L. Maligranda, Some remarks on $s$-convex functions, Aequationes Math., 48 (1994), 100–111. https://doi.org/10.1007/BF01837981 doi: 10.1007/BF01837981
[14]	M. A. Khan, Y. M. Chu, T. U. Khan, J. Khan, Some new inequalities of Hermite-Hadamard type for $s$-convex functions with applications, Open Math., 15 (2017), 1414–1430. https://doi.org/10.1515/math-2017-0121 doi: 10.1515/math-2017-0121
[15]	P. Korus, An extension of the Hermite-Hadamard inequality for convex and $s$-convex functions, Aequationes Math., 93 (2019), 527–534. https://doi.org/10.1007/s00010-019-00642-z doi: 10.1007/s00010-019-00642-z
[16]	B. Meftah, A. Lakhdari, D. C. Benchettah, Some new Hermite-Hadamard type integral inequalities for twice differentiable $s$-convex functions, Comput. Math. Model., 33 (2022), 330–353. https://doi.org/10.1007/s10598-023-09576-3 doi: 10.1007/s10598-023-09576-3
[17]	R. E. Moore, Interval analysis, Prentice-Hall, 1966.
[18]	M. A. Ali, H. Budak, M. Z. Sarikaya, E. Set, Quantum Ostrowski type inequalities for pre-invex functions, Math. Slovaca, 72 (2022), 1489–1500. https://doi.org/10.1515/ms-2022-0101 doi: 10.1515/ms-2022-0101
[19]	M. A. Ali, M. Z. Sarikaya, H. Budak, Z. Y. Zhang, Quantum Hermite-Hadamard type inequalities and related inequalities for subadditive functions, Miskolc Math. Notes, 24 (2023), 5–13. http://doi.org/10.18514/MMN.2023.3769 doi: 10.18514/MMN.2023.3769
[20]	Z. Y. Zhang, M. A. Ali, H. Budak, M. Z. Sarikaya, On Hermite-Hadamard type inequalities for interval-valued multiplicative integrals, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 69 (2020), 1428–1448. https://doi.org/10.31801/cfsuasmas.754842 doi: 10.31801/cfsuasmas.754842
[21]	H. Budak, H. Kara, M. A. Ali, S. Khan, Y. M. Chu, Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions, Open Math., 19 (2021), 1081–1097. https://doi.org/10.1515/math-2021-0067 doi: 10.1515/math-2021-0067
[22]	H. Budak, H. Kara, S. Erden, On Fejer type inclusions for products of interval-valued convex functions, Filomat, 35 (2021), 4937–4955. https://doi.org/10.2298/FIL2114937B doi: 10.2298/FIL2114937B
[23]	T. M. Costa, G. N. Silva, Y. Chalco-Cano, H. Roman-Flores, Gauss-type integral inequalities for interval and fuzzy-interval-valued functions, Comput. Appl. Math., 38 (2019), 58. https://doi.org/10.1007/s40314-019-0836-2 doi: 10.1007/s40314-019-0836-2
[24]	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 doi: 10.1016/j.fss.2017.02.001
[25]	T. M. Costa, H. Roman-Flores, Y. Chalco-Cano, Opial-type inequalities for interval-valued functions, Fuzzy Sets Syst., 358 (2019), 48–63. https://doi.org/10.1016/j.fss.2018.04.012 doi: 10.1016/j.fss.2018.04.012
[26]	T. S. Du, Y. Peng, Hermite-Hadamard type inequalities for multiplicative Riemann-Liouville fractional integrals, J. Comput. Appl. Math., 440 (2024), 115582. https://doi.org/10.1016/j.cam.2023.115582 doi: 10.1016/j.cam.2023.115582
[27]	T. S. Du, S. H. Wu, S. J. Zhao, M. U. Awan, Riemann-Liouville fractional Hermite-Hadamard inequalities for h-preinvex functions, J. Comput. Anal. Appl., 25 (2018), 364–384.
[28]	M. B. Khan, H. M. Srivastava, P. O. Mohammed, K. Nonlaopon, Y. S. Hamed, Some new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions, AIMS Math., 7 (2022), 4338–4358. https://doi.org/10.3934/math.2022241 doi: 10.3934/math.2022241
[29]	M. B. Khan, S. Treanta, H. Alrweili, T. Saeed, M. S. Soliman, Some new Riemann-Liouville fractional integral inequalities for interval-valued mappings, AIMS Math., 7 (2022), 15659–15679. https://doi.org/10.3934/math.2022857 doi: 10.3934/math.2022857
[30]	M. Z. Sarikaya, H. Budak, Generalized Hermite-Hadamard type integral inequalities for fractional integrals, Filomat, 30 (2016), 1315–1326. https://doi.org/10.2298/FIL1605315S doi: 10.2298/FIL1605315S
[31]	M. Z. Sarikaya, H. Budak, F. Usta, Some generalized integral inequalities via fractional integrals, Acta Math. Univ. Comenian, 89 (2020), 27–38.
[32]	M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 57 (2013), 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048 doi: 10.1016/j.mcm.2011.12.048
[33]	D. F. Zhao, T. Q. An, G. J. 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 doi: 10.1016/j.fss.2019.10.006
[34]	D. F. Zhao, T. Q. An, G. J. Ye, D. F. M. Torres, On Hermite-Hadamard type inequalities for harmonical $h$-convex interval-valued functions, Math. Inequal. Appl., 23 (2020), 95–105. http://doi.org/10.7153/mia-2020-23-08 doi: 10.7153/mia-2020-23-08
[35]	D. F. Zhao, G. J. Ye, W. Liu, D. F. M. Torres, Some inequalities for interval-valued functions on time scales, Soft Comput., 23 (2019), 6005–6015. https://doi.org/10.1007/s00500-018-3538-6 doi: 10.1007/s00500-018-3538-6
[36]	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 doi: 10.1090/proc/14741
[37]	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 doi: 10.1016/j.fss.2014.04.005
[38]	H. Budak, H. Kara, F. Hezenci, Generalized Hermite-Hadamard inclusions for a generalized fractional integral, Rocky Mountain J. Math., 53 (2023), 383–395. https://doi.org/10.1216/rmj.2023.53.383 doi: 10.1216/rmj.2023.53.383
[39]	D. F. Zhao, M. A. Ali, A. Kashuri, H. Budak, M. Z. Sarikaya, Hermite-Hadamard-type inequalities for the interval-valued approximately $h$-convex functions via generalized fractional integrals, J. Inequal. Appl., 2020 (2020), 1–38. https://doi.org/10.1186/s13660-020-02488-5 doi: 10.1186/s13660-020-02488-5
[40]	R. E. Moore, Methods and applications of interval analysis, Philadelphia: SIAM, 1979.
[41]	W. W. Breckner, Continuity of generalized convex and generalized concave set-valued functions, Rev. Anal. Numer. Theor. Approx., 22 (1993), 39–51.
[42]	I. Podlubny, Fractional differential equations, Academic Press, 1999.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)