Abstract
In this paper, we have established some generalized Simpson type integral inequalities for generalized fractional integral. The results presented here would provide some fractional inequalities involving k-fractional integral and Riemann–Liouville type fractional operators.
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Alomari, M., Darus, M., Dragomir, S.S.: New inequalities of Simpsonís type for \(s\)-convex functions with applications. RGMIA Res Rep Coll 12(4); (2009) (Article 9)
Chen, J., Huang, X.: Some new inequalities of Simpson’s type for \(s\)-convex functions via fractional integrals. Filomat 31(15), 4989–4997 (2017)
Dragomir, S.S., Agarwal, R.P., Cerone, P.: On Simpsonís inequality and applications. J. Inequal. Appl. 5, 533–579 (2000)
Dragomir, S.S.: On Simpson’s quadrature formula for differentiable mappings whose derivatives belong to \(l_{p}\) spaces and applications. J. KSIAM 2, 57–65 (1998)
Dragomir, S.S.: On Simpson’s quadrature formula for Lipschitzian mappings and applications Soochow. J. Math. 25, 175–180 (1999)
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, 358–369 (2017)
Hussain, S., Qaisar, S.: More results on Simpson’s type inequality through convexity for twice differentiable continuous mappings, vol. 5. Springer, Berlin, pp 77 (2016)
Liu, B.Z.: An inequality of Simpson type. Proc. R. Soc. A 461, 2155–2158 (2005)
Mubeen, S., Habibullah, G.M.: k-Fractional integrals and application. Int. J. Contemp. Math. Sci. 7(2), 89–94 (2012)
Pecaric, J., Proschan, F., Tong, Y.L.: Convex functions, partial ordering and statistical applications. Academic Press, New York (1991)
Pecaric, J., Varosanec, S.: A note on Simpson’s inequality for functions of bounded variation. Tamkang J. Math. 31(3), 239–242 (2000)
Qaisar, S., He, C.J., Hussain, S.: A generalizations of Simpson’s type inequality for differentiable functions using \((\alpha ,m) \)-convex functions and applications. J. Inequal. Appl. 13 (2013) (Article 158)
Kavurmaci, H., Akdemir, A.O., Set, E., Sarikaya, M.Z.: Simpson’s type inequalities for \(m\)-and \((\alpha, m)\) -geometrically convex functions. Konuralp J. Math. 2(1), 90–101 (2014)
Ozdemir, M.E., Akdemir, A.O., Kavurmacı, H.: On the Simpson’s inequality for convex functions on the co-ordinates. Turk. J. Anal. Number Theory 2(5), 165–169 (2014)
Sarıkaya, M. Z., Ertuğral, F.: On the generalized Hermite-Hadamard inequalities 2017 (submitted)
Sarikaya, M.Z., Set, E., Ozdemir, M.E.: On new inequalities of Simpson’s type for \(s\)-convex functions. Comput. Math. Appl. 60, 2191–2199 (2010)
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) (Article 2)
Sarikaya, M.Z., Set, E., Ozdemir, M.E.: On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex. J. Appl. Math. Stat. Inf. 9(1) (2013)
Sarıkaya, M.Z., Tunç, T., Budak, H.: Simpson’s type inequality for \(F\)-convex function. Facta Universitatis Ser. Math. Inform. (in press)
Set, E., Ozdemir, M.E., Sarikaya, M.Z.: On new inequalities of Simpson’s type for quasi-convex functions with applications. Tamkang J. Math. 43(3), 357–364 (2012)
Set, E., Sarikaya, M. Z., Uygun, N.: On new inequalities of Simpson’s type for generalized quasi-convex functions. Adv. Inequal. Appl. 3, 1–11 (2017)
Tseng, K.L., Yang, G.S., Dragomir, S.S.: On weighted Simpson type inequalities and applications. J. Math. Inequal. 1(1), 13–22 (2007)
Ujevic, N.: Double integral inequalities of Simpson type and applications. J. Appl. Math. Comput. 14(1–2), 213–223 (2004)
Yang, Z.Q., Li, Y.J., Du, T.: A generalization of Simpson type inequality via differentiable functions using (\(s, m\) )-convex functions. Ital. J. Pure Appl. Math. 35, 327–338 (2015)
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Ertuğral, F., Sarikaya, M.Z. Simpson type integral inequalities for generalized fractional integral. RACSAM 113, 3115–3124 (2019). https://doi.org/10.1007/s13398-019-00680-x
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DOI: https://doi.org/10.1007/s13398-019-00680-x