Document Type : Research Paper
Authors
1 Department of Mathematics, Faculty of Science and Arts, Duzce University, 81620, Duzce, Turkiye.
2 Department of Mathematics, Faculty of Science and Arts, Duzce University, 81620, Duzce, Turkiye
Abstract
In the literature, several papers are devoted to inequalities of Simpson-type in the case of differentiable convex functions and fractional versions. Moreover, some papers are focused on inequalities of Simpson-type for twice differentiable convex functions. In this research article, we obtain an identity for twice differentiable convex functions. Then, we prove several fractional inequalities of Simpson-type for convex functions.
Keywords
- Simpson type inequalities
- Twice differentiable convex functions
- Riemann-Liouville fractional integrals
- Fractional calculus
Main Subjects
[1] T. Abdeljawad, S. Rashid, Z. Hammouch, I. Iscan and Y.M. Chu, Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications, Adv. Difference Equ., 2020 (2020), pp. 1-26.
[2] M. Alomari, M. Darus and S.S. Dragomir, New inequalities of Simpson's type fors-convex functions with applications, RGMIA Res. Rep. Coll., 12 (2009).
[3] H. Budak, S. Erden and M.A. Ali, Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Methods Appl. Sci., 44 (2021), pp. 378-390.
[4] J. Chen and X. Huang, Some new inequalities of Simpson's type for $s$-convex functions via fractional integrals, Filomat, 31 (2017), pp. 4989-4997.
[5] S.S. Dragomir, R.P. Agarwal and P. Cerone, On Simpson's inequality and applications, J. Inequal. Appl., 5 (2000), pp. 533-579.
[6] T. Du, Y. Li and Z. Yang, A generalization of Simpson's inequality via differentiable mapping using extended $(s,m)$-convex functions, Appl. Math. Comput., 293 (2017), pp. 358-369.
[7] F. Ertugral and M.Z. Sarikaya, Simpson type integral inequalities for generalized fractional integral, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 113 (2019), pp. 115-3124.
[8] R. Goreno and F. Mainardi, Fractional calculus: integral and differential equations of fractional order,, Wien: Springer-Verlag, 223-276, 1997.
[9] J. Hua, B.Y. Xi and F. Qi, Some new inequalities of Simpson type for strongly $s$-convex functions, Afr. Mat., 26 (2015), pp. 741-752.
[10] S. Hussain J. Khalid and Y.M. Chu, Some generalized fractional integral Simpson's type inequalities with applications, AIMS Mathematics, 5 (2020), pp. 5859-5883.
[11] S. Hussain and S. Qaisar, More results on Simpson's type inequality through convexity for twice differentiable continuous mappings, SpringerPlus, 5 (2016), pp. 1-9.
[12] M. Iqbal, S. Qaisar and S. Hussain, On Simpson's type inequalities utilizing fractional integrals, J. Comput. Anal. Appl., 23 (2017), pp. 1137-1145.
[13] I. Iscan, Hermite-Hadamard and Simpson-like type inequalities for differentiable harmonically convex functions, J. Math., 2014 (2014).
[14] S. Kermausuor, Simpson's type inequalities via the Katugampola fractional integrals for s-convex functions, Kragujevac J. Math., 45 (2021), pp. 709-720.
[15] 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.
[16] H. Lei, G. Hu, J. Nie and T. Du, Generalized Simpson-type inequalities considering first derivatives through thek-Fractional Integrals, IAENG Int. J. Appl. Math., 50 (2020), pp. 1-8.
[17] Y. Li and T. Du, Some Simpson type integral inequalities for functions whose third derivatives are ($\alpha ,m$)-GA-convex functions, J. Egyptian Math. Soc., 24 (2016), pp. 175-180.
[18] B.Z. Liu, An inequality of Simpson type, Proc. R. Soc. A, 461 (2005), pp. 2155-2158.
[19] W. Liu, Some Simpson type inequalities for $h$-convex and ($\alpha ,m$)-convex functions, J. Comput. Anal. Appl., 16 (2014), pp. 1005-1012.
[20] C. Luo and T. Du, Generalized Simpson type inequalities involving Riemann-Liouville fractional integrals and their applications, Filomat, 34 (2020), pp. 751-760.
[21] M. Matloka, Some inequalities of Simpson type for h-convex functions via fractional integrals, Abstr. Appl. Anal., Article ID 956850, 5 pages, (2015).
[22] S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, NewYork: Wiley, 1993.
[23] M.E. Ozdemir, A.O. Akdemir and H. Kavurmaci, On the Simpson's inequality for convex functions on the coordinates, Turkish Journal of Analysis and Number Theory, 2 (2014), pp. 165-169.
[24] J. Park, On Simpson-like type integral inequalities for differentiable preinvex functions, Appl. Math. Sci., 7 (2013), pp. 6009-6021.
[25] S. Rashid, A.O. Akdemir, F. Jarad, M.A. Noor and K.I. Noor, Simpson's type integral inequalities for $\kappa $ -fractional integrals and their applications, AIMS Mathematics, 4 (2019), pp. 1087-1100.
[26] M.Z. Sarikaya, E. Set and M.E. Ozdemir, On new inequalities of Simpson's type for convex functions, RGMIA Res. Rep. Coll, 13 (2010), Article2.
[27] M.Z. Sarikaya, E. Set and M.E. Ozdemir, On new inequalities of Simpson's type for$s$-convex functions, Comput. Math. Appl., 60 (2020), pp. 2191-2199.
[28] 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. Inform., 9 (2013), pp. 37-45.
[29] M.Z. Sarikaya, H. Budak and S. Erden, On new inequalities of Simpson's type for generalized convex functions, Korean J. Math., 27 (2019), pp. 279-295.
[30] E. Set, A.O. Akdemir and M.E. Ozdemir, Simpson type integral inequalities for convex functions via Riemann-Liouville integrals, Filomat, 31 (2017), pp. 4415-4420.
[31] M. Vivas-Cortez, T. Abdeljawad, P.O. Mohammed and Y. Rangel-Oliveros, Simpson's integral inequalities for twice differentiable convex functions, Math. Probl. Eng., 2020 (2020).
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