# Certain Simpson-type inequalities for twice-differentiable functions by conformable fractional integrals

## Main Article Content

## Abstract

In this paper, an equality is established by twice-differentiable convex functions with respect to the conformable fractional integrals. Moreover, several Simpson-type inequalities are presented for the case of twice-differentiable convex functions via conformable fractional integrals by using the established equality. Furthermore, our results are provided by using special cases of obtained theorems.

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## References

[1] J.E. Pecaric, F. Proschan and Y. L. Tong, Convex functions, partial orderings, and statistical applications, Mathematics in Science and Engineering, 187. Academic Press, Inc., Boston, MA, 1992. Google Scholar

[2] M. Sarikaya, E. Set, and M. Ozdemir, On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex, Journal of Applied Mathematics, Statistics and Informatics 9 (1) (2013), 37–45. Google Scholar

[3] F. Hezenci, H. Budak, and H. Kara, New version of Fractional Simpson type inequalities for twice differentiable functions, Adv. Difference Equ. 2021 (460), 2021. Google Scholar

[4] M.Z. Sarikaya and N. Aktan, On the generalization of some integral inequalities and their applications, Mathematical and computer Modelling 54 (9-10) (2011), 2175–2182. Google Scholar

[5] D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculus: Models and numerical methods, World Scientific: Singapore, 2016. Google Scholar

[6] G.A. Anastassiou, Generalized fractional calculus: New advancements and applications, Springer: Switzerland, 2021. Google Scholar

[7] N. Iqbal, A. Akgul, R. Shah, A. Bariq, M.M. Al-Sawalha, A. Ali, On Solutions of Fractional- Order Gas Dynamics Equation by Effective Techniques, J. Funct. Spaces Appl. 2022 (2022), 3341754. Google Scholar

[8] N. Attia, A. Akgul, D. Seba, A. Nour, An efficient numerical technique for a biological population model of fractional order, Chaos, Solutions & Fractals 141 (2020), 110349. Google Scholar

[9] A. Gabr, A.H. Abdel Kader, M.S. Abdel Latif, The Effect of the Parameters of the Generalized Fractional Derivatives On the Behavior of Linear Electrical Circuits, International Journal of Applied and Computational Mathematics 7 (2021), 247. Google Scholar

[10] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70. Google Scholar

[11] 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. Google Scholar

[12] D. Zhao and M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo 54 (2017), 903–917. Google Scholar

[13] A. Hyder, A. H. Soliman, A new generalized θ-conformable calculus and its applications in mathematical physics, Phys. Scr. 96 (2020), 015208. Google Scholar

[14] A.A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Dif- ferential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006. Google Scholar

[15] F. Jarad, E. Ugurlu, T. Abdeljawad, and D. Baleanu, On a new class of fractional operators, Adv. Difference Equ. 2017 (2017), 247. Google Scholar

[16] X. You, F. Hezenci, H. Budak, and H. Kara, New Simpson type inequalities for twice differentiable functions via generalized fractional integrals, AIMS Mathematics 7 (3) (2021), 3959–3971. Google Scholar

[17] 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 (17) (2021), 12522–12536. Google Scholar

[18] 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 (3), (2022), 607–616. Google Scholar

[19] T. Abdeljawad, On conformable fractional calculus J. Comput. Appl. Math. 279 (2015), 57–66. Google Scholar