Document Type : Research Paper
Author
Department of Mathematics, Faculty of Science, University of Ordu, Ordu, Turkey.
Abstract
In this paper, it is aimed to improvement the boundaries of fractional integral operators containing extended generalized Mittag-Leffler functions. The offered results enhance the previously known bounds of the distinct fractional integral operators for strongly $p$-convex functions of higher order. The acquired inequalities ensure boundedness, continuity, and Hadamard type inequality for fractional integrals containing an expanded Mittag-Leffler function. Furthermore, these results can be refined for various classes of strongly convex functions offered in the literature.
Keywords
- Mittag-Leffler function
- Strongly $p$-convexity of higher order
- $k$-fractional integrals
- Hadamard inequality
- Fej\'er-Hadamard inequality
Main Subjects
1. H. Angulo, J. Giménez, A.M. Moros and K. Nikodem, On strongly h-convex functions, Ann. Funct. Anal., 2 (2011), pp. 85-91.
2. H. Budak, On Fejer type inequalities for convex mappings utilizing fractional integrals of a function with respect to another function, Result. Math., 74 (2019), pp. 1-15.
3. H. Budak, S. Kılınç Yıldırım, M.Z. Sarıkaya and H. Yıldırım, Some parameterized Simpson-, midpoint and trapezoid-type inequalities for generalized fractional integrals, J. Inequal. Appl., 40 (2022).
4. L. Chen, G. Farid, S.I. Butt and S.B. Akbar, Boundedness of fractional integral operators containing Mittag-Leffler functions, Turkish J. Ineq., 4 (2020), pp. 14-24.
5. Z. Chen, G. Farid, A.U. Rehman and N. Latif, Estimations of fractional integral operators for convex functions and related results, Adv. Difference Equ., 1 (2020), pp. 1-18.
6. Z. Chen, G. Farid, M. Saddiqa, S. Ullah and N. Latif, Study of fractional integral inequalities involving Mittag-Leffler functions via convexity, J. Inequal. Appl., 1 (2020), pp. 206.
7. Y. Dong, M. Saddiqa, S. Ullah and G. Farid, Study of fractional integral operators containing Mittag-Leffler functions via strongly (α,m)-convex functions, Math. Probl. Eng., Article ID 6693914, (2021), pp. 13.
8. Y. Erdem, H. Ogünmez and H. Budak, Some generalized inequalities of Hermite-Hadamard type for strongly s-convex functions, NTMSCI, 5 (2017), pp. 22-32.
9. G. Farid, A unified integral operator and further its consequences, Open J. Math. Anal., 4 (2020), pp. 1-7.
10. G. Farid, Some Riemann–Liouville fractional integral inequalities for convex functions, J. Anal., 27 (2019), pp. 1095-1102.
11. G. Farid, S. B. Akbar, S. U. Rehman and J. Pecarić, Boundedness of fractional integral operators containing Mittag-Leffler functions via (s,m)-convexity, AIMS Math., 5 (2020), pp. 966-978.
12. R. Ger and K. Nikodem, Strongly convex functions of higher order, Nonlinear Anal., Theory Methods Appl., 74 (2011), pp. 661-665.
13. G. Hong, G. Farid, W. Nazeer, S.B. Akbar, J. Pecarić, J. Zou and S. Geng, Boundedness of fractional integral operators containing Mittag-Leffler function via exponentially s-convex functions, J. Math., Article ID 3584105, (2020), pp. 7.
14. A. Hassan, A.R. Khan, N. Irshad and S. Khatoon, Fractional Ostrowski-type inequalities via (α, β, γ, δ)-convex functions, Sahand Commun. Math. Anal., 20 (2023), pp. 1-20.
15. Í. Íşcan, Ostrowski type inequalities for p-convex functions, NTMSCI, 4 (2016), pp. 140-150.
16. Y.C. Kwun, G. Farid, S. Ullah, W. Nazeer, K. Mahreen and S.M. Kang, Inequalities for a unified integral operator and associated results in fractional calculus, IEEE Access, 7 (2019), pp. 126283-126292.
17. C. Karim and E. Maulana, A note on generalized strongly p-convex functions of higher order, CAUCHY–Jurnal Matematika Murni dan Aplikasi, 45 (2014), pp. 89-94.
18. G.H. Lin and M. Fukushima, Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 118 (2003), pp. 67-80.
19. B.B. Mohsin, M.Z. Javed, M.U. Awan, M.V. Mihai, H. Budak, A.G. Khan and M.A. Noor, Jensen-Mercer type inequalities in the setting of fractional calculus with applications, Symmetry, 14 (2022), pp. 2187.
20. B. Meftah and C. Marrouche, Ostrowski type inequalities for ntimes strongly m−MT-convex functions, Sahand Commun. Math. Anal., 20 (2023), pp. 81-96.
21. B.B. Mohsen, M.A. Noor, K.I. Noor and M. Postolache, Strongly convex functions of higher order involving bifunction, Mathematics, 7 (2019), pp. 1028.
22. B. Ni, G. Farid and K. Mahreen, Inequalities for a unified integral operator via (α,m)-convex functions, J. Math., Article ID 2345416, (2020), pp. 9.
23. K. Nonlaopon, G. Farid, H. Yasmeen, F.A. Shah and C.Y. Jung, Generalization of some fractional integral operator inequalities for convex functions via unified Mittag-Leffler function, Symmetry, 14 (2022), pp. 922.
24. M.A. Noor, K.I. Noor and S. Iftikhar, Hermite-Hadamard inequalities for strongly harmonic convex functions, J. Inequal. Spec. Funct., 7 (2016), pp. 99-113.
25. X. Qiang, G. Farid, J. Pecarić and S.B. Akbar, Generalized fractional integral inequalities for exponentially (s,m)-convex functions, J. Inequal. Appl., 70 (2020).
26. G. Rahman, T. Abdeljawad, F. Jarad and K.S. Nisar, Bounds of generalized proportional fractional integrals in general form via convex functions and their applications, Mathematics, 8 (2020), pp. 113.
27. S. Rashid, M.A. Noor, K.I. Noor and Y. Chu, Ostrowski type inequalities in the sense of generalized K-fractional integral operator for exponentially convex functions, AIMS Math., 5 (2020), pp. 2629-2645.
28. M.S. Saleem, Y.M. Chu, N. Jahangir, H. Akhtar and C.Y. Jung, On generalized strongly p-convex functions of higher order, J. Math., Article ID 8381431, (2020), pp. 8.
29. E. Set, S.I. Butt, A.O. Akdemir, A. Karaoglan and T. Abdeljawad, New integral inequalities for differentiable convex functions via Atangana-Baleanu fractional integral operators, Chaos Solitons Fractals, 143 (2021), 110554.
30. S. Turhan, A.K. Demirel, S. Maden and Í. Íşcan, Hermite-Hadamard inequality for strongly p-convex functions, Turk. J. Math. Comput. Sci., 10 (2018), pp. 184-189.
31. Y. Yue, G. Farid, A.K. Demirel, W. Nazeer and Y. Zhao, Hadamard and Fejér-Hadamard inequalities for generalized k-fractional integrals involving further extension of Mittag-Leffler functions, AIMS Math., 7 (2022), pp. 681-703.
32. M. Yussouf, G. Farid, K.A. Khan and C.Y. Jung, Hadamard and Fejér-Hadamard inequalities for further generalized fractional integrals involving Mittag-Leffler functions, J. Math., Article ID 5589405, (2021), pp. 13.
33. C.T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008), pp. 266-272.
34. G. Teschl, Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators, Graduate Studies in Mathematics, Amer. Math. Soc., Rhode Island, 2009.
35. J. Li, M. Yasuda and J. Song, Regularity properties of null-additive fuzzy measure on metric space, in: Proc. 2nd Internatinal Conference on Modeling Decisions for Artificial Intelligencer, Tsukuba, Japan, 2005, 59-66.
)