Inequalities of Simpson-type for twice-differentiable convex functions via conformable fractional integrals

Document Type : Research Paper


Department of Mathematics, Faculty of Science and Arts, Duzce University, Turkiye


This paper proves an equality for the case of twice-differentiable convex functions involving conformable fractional integrals. Using the established equality, we give new Simpson-type inequalities for the case of twice-differentiable convex functions via conformable fractional integrals. We also consider some special cases which can be deduced from the main results.


[1] A.A. Abdelhakim, The flaw in the conformable calculus: It is conformable because it is not fractional, Fract. Calc. Appl. Anal. 22 (2019), 242–254.
[2] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015), 57–66.
[3] G.A. Anastassiou, Generalized Fractional Calculus: New Advancements and Applications, Springer, Switzerland, 2021.
[4] N. Attia, A. Akgul, D. Seba, and A. Nour, An efficient numerical technique for a biological population model of fractional order, Chaos Solitons Fractals 141 (2020), 110349.
[5] D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, Singapore, 2016.
[6] H. Budak, H. Kara, and F. Hezenci, Fractional Simpson-type inequalities for twice differentiable functions, Sahand Commun. Math. Anal. 20 (2023), no. 3, 97–108.
[7] H. Desalegn, J.B. Mijena, E.R. Nwaeze, and T. Abdi, Simpson’s type inequalities for s-convex functions via generalized proportional fractional integral, Foundations 2 (2022), no. 3, 607–616.
[8] A. Gabr, A.H. Abdel Kader, and M.S. Abdel Latif, The effect of the parameters of the generalized fractional derivatives on the behavior of linear electrical circuits, Int. J. Appl. Comput. Math. 7 (2021), 247.
[9] N. Iqbal, A. Akgul, R. Shah, A. Bariq, M.M. Al-Sawalha, and A. Ali, On solutions of fractional-order gas dynamics equation by effective techniques, J. Funct. Spaces 2022 (2022), 3341754.
[10] F. Hezenci, H. Budak, and H. Kara, New version of fractional Simpson type inequalities for twice differentiable functions, Adv. Difference Equ. 2021 (2021), no. 1, 1–10.
[11] H. Budak, F. Hezenci, and H. Kara, On parameterized inequalities of Ostrowski and Simpson type for convex functions via generalized fractional integrals, Math. Methods Appl. Sci. 44 (2021), no. 17, 12522–12536.
[12] A. Hyder and A.H. Soliman, A new generalized θ-conformable calculus and its applications in mathematical physics, Physica Scripta 96 (2020), 015208.
[13] F. Jarad, E. Ugurlu, T. Abdeljawad, and D. Baleanu, On a new class of fractional operators, Adv. Difference Equ. 2017 (2017), 247.
[14] U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), 1–15.
[15] S. Kermausuor, Simpson’s type inequalities via the Katugampola fractional integrals for s-convex functions, Kragujev. J. Math. 45 (2021), 709–720.
[16] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70.
[17] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
[18] M.Z. Sarikaya and N. Aktan, On the generalization of some integral inequalities and their applications, Math. Comput. Modell. 54 (2011), no. 9-10, 2175–2182.
[19] M.Z. Sarikaya, E. Set, and M.E. Ozdemir, On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex, J. Appl. Math. Stat. Inf. 9 (2013), no. 1, 37–45.
[20] D. Zhao and M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo 54 (2017), 903–917.
Volume 15, Issue 3
March 2024
Pages 1-10
  • Receive Date: 29 September 2022
  • Accept Date: 19 October 2023