Reverse fractional integral inclusions and generic $ \eta_{\hslash} $ interval-valued convexity

Muhammad Samraiz; Somia Zafar; Muath Awadalla; Hajer Zaway; Muhammad Samraiz; Somia Zafar; Muath Awadalla; Hajer Zaway

doi:10.3934/math.2025725

AIMS Mathematics

2025, Volume 10, Issue 7: 16200-16232. doi: 10.3934/math.2025725

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Research article

Reverse fractional integral inclusions and generic $ \eta_{\hslash} $ interval-valued convexity

1.
Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan; Email: somiazafar07@gmail.com (S.Z)
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia; Email: mawadalla@kfu.edu.sa (M.A)

Received: 30 April 2025 Revised: 30 June 2025 Accepted: 09 July 2025 Published: 17 July 2025
MSC : 26A33, 26A51, 26D10, 26E25

This paper presents the development of reverse Minkowski and reverse Hölder's fractional integral inclusions. We propose a generic class of $ \eta_{\hslash} $ interval-valued $ (\mathrm{I}.\mathrm{V}) $ convex functions, which unifies various existing classes. Additionally, we obtain a discrete Jensen-type inclusion within this convexity setup. By leveraging this advanced convexity structure together with tempered fractional integral operators, we derive new Hermite–Hadamard ($ \mathrm{H} $-$ \mathrm{H} $)-type, Fejér-$ \mathrm{H} $-$ \mathrm{H} $-type, and other fractional inclusions. Moreover, we explore the broader significance of our results, supporting them with graphical visualizations. The applications of our results are demonstrated through average value computations.
Citation: Muhammad Samraiz, Somia Zafar, Muath Awadalla, Hajer Zaway. Reverse fractional integral inclusions and generic $ \eta_{\hslash} $ interval-valued convexity[J]. AIMS Mathematics, 2025, 10(7): 16200-16232. doi: 10.3934/math.2025725

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Abstract

This paper presents the development of reverse Minkowski and reverse Hölder's fractional integral inclusions. We propose a generic class of $ \eta_{\hslash} $ interval-valued $ (\mathrm{I}.\mathrm{V}) $ convex functions, which unifies various existing classes. Additionally, we obtain a discrete Jensen-type inclusion within this convexity setup. By leveraging this advanced convexity structure together with tempered fractional integral operators, we derive new Hermite–Hadamard ($ \mathrm{H} $-$ \mathrm{H} $)-type, Fejér-$ \mathrm{H} $-$ \mathrm{H} $-type, and other fractional inclusions. Moreover, we explore the broader significance of our results, supporting them with graphical visualizations. The applications of our results are demonstrated through average value computations.

References

[1]	S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Science Direct Working Paper, 2003.
[2]	J. E. Peajcariaac, Y. L. Tong, Convex functions, partial orderings and statistical applications, Academic Press, 1992.
[3]	S. Wu, M. U. Awan, M. A. Noor, K. I. Noor, S. Iftikhar, On a new class of convex functions and integral inequalities, J. Inequal. Appl., 2019 (2019), 131. https://doi.org/10.1186/s13660-019-2074-y doi: 10.1186/s13660-019-2074-y
[4]	R. E. Moore, Interval analysis, Englewood Cliffs: Prentice-Hall, 1966.
[5]	J. P. Merlet, Interval analysis for certified numerical solution of problems in robotics, Int. J. Appl. Math. Comput. Sci., 19 (2009), 399–412. https://doi.org/10.2478/v10006-009-0033-3 doi: 10.2478/v10006-009-0033-3
[6]	E. J. Rothwell, M. J. Cloud, Automatic error analysis using intervals, IEEE T. Educ., 55 (2011), 9–15. https://doi.org/10.1109/TE.2011.2109722 doi: 10.1109/TE.2011.2109722
[7]	J. M. Snyder, Interval analysis for computer graphics, In: Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, 1992,121–130.
[8]	O. Strauss, A. Rico, Towards interval-based non-additive deconvolution in signal processing, Soft Comput., 16 (2012), 809–820. https://doi.org/10.1007/s00500-011-0771-7 doi: 10.1007/s00500-011-0771-7
[9]	H. T. Nguyen, V. Kreinovich, Q. Zuo, Interval-valued degrees of belief: Applications of interval computations to expert systems and intelligent control, Int. J. Uncertain. Fuzz., 5 (1997), 317–358. https://doi.org/10.1142/S0218488597000257 doi: 10.1142/S0218488597000257
[10]	D. Ghosh, A. K. Debnath, W. Pedrycz, A variable and a fixed ordering of intervals and their application in optimization with interval-valued functions, Int. J. Approx. Reason., 121 (2020), 187–205. https://doi.org/10.1016/j.ijar.2020.03.004 doi: 10.1016/j.ijar.2020.03.004
[11]	E. de Weerdt, Q. P. Chu, J. A. Mulder, Neural network output optimization using interval analysis, IEEE T. Neural Networ., 20 (2009), 638–653. https://doi.org/10.1109/TNN.2008.2011267 doi: 10.1109/TNN.2008.2011267
[12]	W. W. Breckner, Continuity of generalized convex and generalized concave set-valued functions, Rev. Anal. Numer. Theor. Approx., 22 (1993), 39–51.
[13]	E. Sadowska, Hadamard inequality and a refinement of Jensen inequality for set-valued functions, Results Math., 32 (1997), 332–337. https://doi.org/10.1007/BF03322144 doi: 10.1007/BF03322144
[14]	R. E. Moore, R. B. Kearfott, M. J. Cloud, Introduction to interval analysis, Society for Industrial and Applied Mathematics, 2009.
[15]	S. G. Samko, Fractional integrals and derivatives: Theory and applications, Gordon and Breach Science Publishers, Switzerland, 1993.
[16]	M. Samraiz, Z. Perveen, T. Abdeljawad, S. Iqbal, S. Naheed, On certain fractional calculus operators and applications in mathematical physics, Phys. Scr., 95 (2020), 115210. https://doi.org/10.1088/1402-4896/abbe4e doi: 10.1088/1402-4896/abbe4e
[17]	R. G. Buschman, Decomposition of an integral operator by use of Mikusiński calculus, SIAM J. Math. Anal., 3 (1972), 83–85. https://doi.org/10.1137/0503010 doi: 10.1137/0503010
[18]	C. Li, C. Deng, L. Zhao, Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discrete Cont. Dyn.-B, 24 (2019), 1989–2015. https://doi.org/10.3934/dcdsb.2019026 doi: 10.3934/dcdsb.2019026
[19]	M. M. Meerschaert, F. Sabzikar, J. Chen, Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14. https://doi.org/10.1016/j.jcp.2014.04.024
[20]	P. O. Mohammed, M. Z. Sarikaya, D. Baleanu, On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry, 12 (2020), 595. https://doi.org/10.3390/sym12040595 doi: 10.3390/sym12040595
[21]	H. Román-Flores, Y. Chalco-Cano, W. Lodwick, Some integral inequalities for interval-valued functions, Comput. Appl. Math., 37 (2018), 1306–1318. https://doi.org/10.1007/s40314-016-0396-7 doi: 10.1007/s40314-016-0396-7
[22]	D. 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
[23]	H. Budak, T. Tunç, M. Sarikaya, Fractional Hermite-Hadamard-type inequalities for interval-valued functions, Proc. Am. Math. Soc., 148 (2020), 705–718. https://doi.org/10.1090/proc/14741 doi: 10.1090/proc/14741
[24]	Z. Li, M. S. Zahoor, H. Akhtar, Hermite-Hadamard and fractional integral inclusions for interval-valued generalized p-convex function, J. Math., 2020 (2020), 4606439. https://doi.org/10.1155/2020/4606439 doi: 10.1155/2020/4606439
[25]	F. Shi, G. Ye, D. Zhao, W. Liu, Some fractional Hermite-Hadamard-type inequalities for interval-valued coordinated functions, Adv. Differ. Equ., 2021 (2021), 1–17. https://doi.org/10.1186/s13662-020-03200-z doi: 10.1186/s13662-020-03200-z
[26]	Z. Sha, G. Ye, D. Zhao, W. Liu, On interval-valued K-Riemann integral and Hermite-Hadamard type inequalities, AIMS Math., 6 (2021), 1276–1295. https://doi.org/10.3934/math.2021079 doi: 10.3934/math.2021079
[27]	M. A. Alqudah, A. Kashuri, P. O. Mohammed, M. Raees, T. Abdeljawad, M. Anwar, et al., On modified convex interval valued functions and related inequalities via the interval valued generalized fractional integrals in extended interval space, AIMS Math., 6 (2021), 4638–4663. https://doi.org/10.3934/math.2021273 doi: 10.3934/math.2021273
[28]	L. Chen, M. S. Saleem, M. S. Zahoor, R. Bano, Some inequalities related to interval-valued $\eta_{h}$-convex functions, J. Math., 2021 (2021), 6617074. https://doi.org/10.1155/2021/6617074 doi: 10.1155/2021/6617074
[29]	B. Bin-Mohsin, M. Z. Javed, M. U. Awan, B. Meftah, A. Kashuri, Fractional reverse inequalities involving generic interval-valued convex functions and applications, Fractal Fract., 8 (2024), 587. https://doi.org/10.3390/fractalfract8100587 doi: 10.3390/fractalfract8100587

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