On inequalities of Simpson's type for convex functions via generalized fractional integrals

Abstract

Fractional calculus and applications have application areas in many different fields such as physics, chemistry, and engineering as well as mathematics. The application of arithmetic carried out in classical analysis in fractional analysis is very important in terms of obtaining more realistic results in the solution of many problems. In this study, we prove an identity involving generalized fractional integrals by using differentiable functions. By utilizing this identity, we obtain several Simpson’s type inequalities for the functions whose derivatives in absolute value are convex. Finally, we present some new results as the special cases of our main results.

Keywords

• Simpson type inequalities
• convex function
• generalized fractional integrals

References

1. Agarwal, P., Vivas-Cortez, M., Rangel-Oliveros, Y., Ali, M. A., New Ostrowski type inequalities for generalized s-convex functions with applications to some special means of real numbers and to midpoint formula, AIMS Mathematics, 7(1) (2022), 1429–1444. doi:10.3934/math.2022084
2. Ali, M. A., Budak, H., Zhang, Z., Yildirim, H., Some new Simpson’s type inequalities for co-ordinated convex functions in quantum calculus, Math. Methods Appl. Sci., 44(6) (2021), 4515–4540 https://doi.org/10.1002/mma.7048
3. Ali, M. A., Budak, H., Abbas, M., Chu, Y.-M., Quantum Hermite–Hadamard-type inequalities for functions with convex absolute values of second qκ2 -derivatives, Adv. Difference Equ., 2021(7) (2021). https://doi.org/10.1186/s13662-020-03163-1
4. Ali, M. A., Chu, Y. M., Budak, H., Akkurt, A., Yildirim, H., New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions, Adv. Difference Equ., 2021(64) (2021). https://doi.org/10.1186/s13662-021-03226-x
5. Ali, M. A., Abbas, M., Budak, H., Agarwal, P., Murtaza, G., Chu, Y.-M., Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables, Adv. Difference Equ., 2021(25) (2021). https://doi.org/10.1186/s13662-020-03195-7
6. Alomari, M., Darus, M., Dragomir, S. S., New inequalities of Simpson’s type for s-convex functions with applications, RGMIA Res. Rep. Coll., 12(4) (2009).
7. Budak, H., Erden, S., Ali, M. A., Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Methods Appl. Sci., 44(1) (2021), 378–390 https://doi.org/10.1002/mma.6742.
8. Budak, H., Kara, H., Kapucu, R., New midpoint type inequalities for generalized fractional integral, Comput. Methods Differ. Equ., 10(1) (2022), 93–108. DOI:10.22034/cmde.2020.40684.1772

9. Budak, H., Pehlivan E., Kösem, P., On new extensions of Hermite-Hadamard inequalities for generalized fractional integrals, Sahand Communications in Mathematical Analysis, 18(1) (2021), 73–88.
10. Budak, H., Hezenci, F., Kara, H., On parameterized inequalities of Ostrowski and Simpson type for convex functions via generalized fractional integrals, Math. Methods Appl. Sci., 44(17) (2021), 12522–12536. DOI: 10.1002/mma.7558
11. Butt, S. I., Akdemir, A. O., Agarwal, P., Baleanu, D., Non-conformable integral inequalitiesof Chebyshev-Polya-Szego type, J. Math. Inequal., 15(4) (2021), 1391–1400. dx.doi.org/10.7153/jmi-2021-15-94
12. Butt, S. I., Agarwal, P., Yousaf, S. Guirao, J. L., Generalized fractal Jensen and Jensen-Mercer inequalities for harmonic convex function with applications, J. Inequal. Appl., 2022(1) (2022), 1-18. https://doi.org/10.1186/s13660-021-02735-3
13. Butt, S. I., Akdemir, A. O., Nadeem, M., Raza, M. A., Gruss type inequalities via generalized fractional operators Math. Methods Appl. Sci., 44(17) (2021), 12559–12574. https://doi.org/10.1002/mma.7563
14. Butt, S. I., Yousaf, S., Akdemir, A. O., Dokuyucu, M. A., New Hadamard-type integral inequalities via a general form of fractional integral operators, Chaos Solitons Fractals, 148 (2021), 111025. https://doi.org/10.1016/j.chaos.2021.111025
15. Butt, S. I., Nadeem, M., Tariq, M., Aslam, A., New integral type inequalities via Rainaconvex functions and its applications Commun. Fac. Sci. Univ. Ank. S´er. A1 Math. Stat., 70(2) (2021), 1011-1035. https://doi.org/10.31801/cfsuasmas.848853
16. Dragomir, S. S., Agarwal, R. P., Cerone, P., On Simpson’s inequality and applications, J. Inequal. Appl., 5 (2000), 533–579.
17. Du, T., Li, Y., Yang, Z., A generalization of Simpson’s inequality via differentiable mapping using extended (s,m)-convex functions, Appl. Math. Comput., 293 (2017), 358–369. https://doi.org/10.1016/j.amc.2016.08.045
18. Erden, S., Iftikhar, S., Delavar, R. M., Kumam, P., Thounthong P., Kumam, W., On generalizations of some inequalities for convex functions via quantum integrals, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 114(3) (2020), 1–15. Doi: 10.1007/s13398-020-00841-3.
19. Ertugral, F., Sarikaya, M. Z., Simpson type integral inequalities for generalized fractional integral, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 113(4) (2019), 3115–3124. https://doi.org/10.1007/s13398-019-00680-x
20. Farid, G., Rehman, A., Zahra, M., On Hadamard inequalities for k-fractional integrals, Nonlinear Funct. Anal. Appl., 21(3) (2016), 463–478.
21. Gorenflo, R., Mainardi, F., Fractional Calculus: Integral and Differential Equations of Fractional Order, Wien: Springer-Verlag, 1997, 223–276.
22. Hadamard, J., Etude sur les proprietes des fonctions entieres et en particulier d’une fonction considree par, Riemann, J. Math. Pures. et Appl., 58 (1893), 171–215.
23. Hai, X., Wang S. H., Simpson type inequalities for convex function based on the generalized fractional integrals, Turkish J. Math., 5(1) (2021), 1–15.
24. Han, J., Mohammed, P. O., Zeng, H., Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function, Open Math., 18(1) (2020), 794–806. https://doi.org/10.1515/math-2020-0038
25. Iftikhar, S., Komam, P., Erden, S., Newton’s type integral inequalities via local fractional integrals, Fractals, 28(3) (2020), 2050037, 13 pages. Doi: 10.1142/S0218348X20500371.
26. Jain, S., Goyal, R., Agarwal, P., Guirao, J. L., Some inequalities of extended hypergeometric functions, Mathematics, 9(21) (2021), 2702. https://doi.org/10.3390/math9212702
27. Katugampola, U. N., A new fractional derivative with classical properties, (2014) e-print arXiv:1410.6535.
28. Kashuri, A., Ali, M. A., Abbas M., Budak, H., New inequalities for generalized m-convex functions via generalized fractional integral operators and their applications, International Journal of Nonlinear Analysis and Applications, 10(2) (2019), 275-299. doi: 10.22075/ijnaa. 2019.18455.2014
29. Kashuri, A., Liko, R., On Fej´er type inequalities for convex mappings utilizing generalized fractional integrals, Appl. Appl. Math., 15(1) (2020), 240–255.
30. Khalil, R., Alomari, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002
31. Miller, S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, New York, Wiley, 1993.
32. Mohammed, P. O., Sarikaya, M. Z., On generalized fractional integral inequalities for twice differentiable convex functions, J. Comput. Appl. Math., 372 (2020), 112740. https://doi.org/10.1016/j.cam.2020.112740
33. Mubeen, S., Habibullah, G. M., k-Fractional integrals and application, Int. J. Contemp. Math. Sciences, 7(2) (2012), 89–94.
34. Noor, M. A., Noor, K. I., Iftikhar, S., Some Newton’s type inequalities for harmonic convex functions, J. Adv. Math. Stud., 9(1) (2016), 07–16.
35. Noor, M. A., Noor K. I., Iftikhar, S., Newton inequalities for p-harmonic convex functions, Honam Math. J., 40(2) (2018), 239–250. https://doi.org/10.5831/HMJ.2018.40.2.239
36. Park, J., On Simpson-like type integral inequalities for differentiable preinvex functions, Appl. Math. Sci., 7(121) (2013), 6009–6021. http://dx.doi.org/10.12988/ams.2013.39498
37. Podlubni, I., Fractional Differential Equations, San Diego, CA: Academic Press, 1999.
38. Sarikaya, M. Z., Set, E., Yaldiz, H., Basak, N., Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 57(9–10) (2013), 2403– 2407. https://doi.org/10.1016/j.mcm.2011.12.048
39. Sarikaya, M. Z., Akkurt, A., Budak, H., Yildirim, M. E., Yildirim, H., Hermite-Hadamard’s inequalities for conformable fractional integrals, Konuralp Journal of Mathematics, 8(2) (2020), 376-383.
40. Sarikaya, M. Z., Ogunmez, H., On new inequalities via Riemann–Liouville fractional integration, Abs. Appl. Anal. 2012. Article ID 428983, 10 pages. doi:10.1155/2012/428983.
41. Sarikaya, M. Z., Ertugral, F., On the generalized Hermite-Hadamard inequalities, An. Univ. Craiova Ser. Mat. Inform., 47(1) (2020), 193–213.
42. Sarikaya, M. Z., Set, E., Özdemir, M. E., On new inequalities of Simpson’s type for convex functions, RGMIA Res. Rep. Coll., 13(2) (2010), Article2.
43. Sarikaya, M. Z., Set, E., Özdemir, M. E., On new inequalities of Simpson’s type for s-convex functions, Comput. Math. Appl., 60(8) (2020), 2191–2199. https://doi.org/10.1016/j.camwa.2010.07.033
44. Set, E., Butt, S. I., Akdemir, A. O., Karaoglan, A., Abdeljawad, T., New integral inequalities for differentiable convex functions via Atangana-Baleanu fractional integral operators, Chaos Solitons Fractals, 143 (2021), 110554. https://doi.org/10.1016/j.chaos.2020.110554
45. Turkay, M. E., Sarikaya, M. Z., Budak, H., Yildirim, H., Some Hermite-Hadamard type inequalities for co-ordinated convex functions via generalized fractional integrals, Submitted, ResearchGate Article: https://www.researchgate.net/publication/321803898.
46. Qi, F., Mohammed, P. O., Yao, J.-C., Yao, Y.-H., Generalized fractional integral inequalities of Hermite–Hadamard type for (α,m)-convex functions, J. Inequal. Appl. 2019, 135 (2019). https://doi.org/10.1186/s13660-019-2079-6
47. Vivas-Cortez, M., Ali, M. A., Kashuri, A., Sial, I. B., Zhang, Z., Some new Newton’s type integral inequalities for co-ordinated convex functions in quantum calculus, Symmetry, 12(9) (2020) 1476; https://doi.org/10.3390/sym12091476.
48. Zhao, D., Ali, M. A., Kashuri, A., Budak, H., Sarikaya, M. Z., Hermite–Hadamard-type inequalities for the interval-valued approximately h−convex functions via generalized fractional integrals, J. Inequal. Appl., 2020(222) (2020), 1–38. https://doi.org/10.1186/s13660-020-02488-5