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A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions

  • Fatih Hezenci EMAIL logo
Published/Copyright: June 15, 2023
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ABSTRACT

In this paper, an equality is proved for twice differentiable convex functions involving Riemann–Liouville fractional integral. With the help of this equality, there are established several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. By using special cases of the main results, previously obtained Simpson type inequalities are found for the Riemann–Liouville fractional integral.

2020 Mathematics Subject Classification: 26D07; 26D10; 26D15

(Communicated by Tomasz Natkaniec)


REFERENCES

[1] Abdeljawad, T.—Rashid, S.—Hammouch, Z.—İşcan, İ.—Chu, Y.-M.: Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications, Adv. Difference Equ. 2020 (2020), Art. No. 496.10.1186/s13662-020-02955-9Search in Google Scholar

[2] Agarwal, P.—Tariboon, J.—Ntouyas, S. K.: Some generalized Riemann–Liouville k-fractional integral inequalities, J. Inequal. Appl. 2016 (2016), Art. No. 122.10.1186/s13660-016-1067-3Search in Google Scholar

[3] Ali, M. A.—Kara, H.—Tariboon, J.—Asawasamrit, S.—Budak, H.—Hezenci, F.: Some new Simpson’s-Formula-Type inequalities for twice-differentiable convex functions via generalized fractional operators, Symmetry 13(12) (2021), Art. No. 2249.10.3390/sym13122249Search in Google Scholar

[4] 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).Search in Google Scholar

[5] 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, 12522–12536.10.1002/mma.7558Search in Google Scholar

[6] Budak, H.—Hezenci, F.—Kara, H.: On generalized Ostrowski, Simpson and trapezoidal type inequalities for co-ordinated convex functions via generalized fractional integrals, Adv. Difference Equ. 2021 (2021), Art. No. 312.10.1186/s13662-021-03463-0Search in Google Scholar

[7] Chen, J.—Huang, X.: Some new inequalities of Simpson’s type for s-convex functions via fractional integrals, Filomat 31(15) (2017), 4989–4997.10.2298/FIL1715989CSearch in Google Scholar

[8] Dragomir, S. S.—Agarwal, R.—Cerone, P. P.: On Simpson’s inequality and applications, J. Inequal. Appl. 5(6) (2000), 533–579.10.1155/S102558340000031XSearch in Google Scholar

[9] Du, T.—Lİ, Y.—Yang, Z.: A generalization of Simpson’s inequality via differentiable mapping using extended (s,m)-convex functions, Appl. Math. Comput. 293 (2017), 358–369.10.1016/j.amc.2016.08.045Search in Google Scholar

[10] Ertugral, F.—Sarikaya, M. Z.: Simpson type integral inequalities for generalized fractional integral, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(4) (2019), 3115–3124.10.1007/s13398-019-00680-xSearch in Google Scholar

[11] Gorenflo, R.—Mainardi, F.: Fractional calculus: Integral and differential equations of fractional order. In: Fractals and Fractional Calculus in Continuum Mechanics (Carpinteri, A., Mainardi, F., eds.), International Centre for Mechanical Sciences, vol. 378, Springer, Vienna, 1997, pp. 223–276.10.1007/978-3-7091-2664-6_5Search in Google Scholar

[12] Hezenci, F.—Budak, H.—Kara, H.: New version of Fractional Simpson type inequalities for twice differentiable functions, Adv. Difference Equ. 2021 (2021), Art. No. 460.10.1186/s13662-021-03615-2Search in Google Scholar

[13] Hussain, S.—Khalid, J.—Chu, Y. M.: Some generalized fractional integral Simpson’s type inequalities with applications, AIMS Math. 5(6) (2020), 5859–5883.10.3934/math.2020375Search in Google Scholar

[14] Hussain, S.—Qaisar, S.: More results on Simpson’s type inequality through convexity for twice differentiable continuous mappings, SpringerPlus 5(1), 2016, 1–9.10.1186/s40064-016-1683-xSearch in Google Scholar PubMed PubMed Central

[15] Iqbal, M.—Qaisar, S.—Hussain, S.: On Simpson’s type inequalities utilizing fractional integrals, J. Comput. Anal. Appl. 23(6) (2017), 1137–1145.Search in Google Scholar

[16] İşcan, İ.: Hermite-Hadamard and Simpson-like type inequalities for differentiable harmonically convex functions, J. Math. 2014 (2014).10.1155/2014/346305Search in Google Scholar

[17] Kermausuor, S.: Simpson’s type inequalities via the Katugampola fractional integrals for s-convex functions, Kragujevac J. Math. 45(5) (2021), 709–720.10.46793/KgJMat2105.709KSearch in Google Scholar

[18] Kilbas, A. A.—Srivastava, H. M.—Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier Sci. B. V., Amsterdam, 2006.Search in Google Scholar

[19] Lei, H.—Hu, G.—Nie, J.—Du, T.: Generalized Simpson-type inequalities considering first derivatives through the k-fractional integrals, IAENG Int. J. Appl. Math. 50(3) (2020), 1–8.Search in Google Scholar

[20] Li, Y.—Du, T.: Some Simpson type integral inequalities for functions whose third derivatives are (α,m)-GA-convex functions, J. Egyptian Math. Soc. 24(2) (2016), 175–180.10.1016/j.joems.2015.05.009Search in Google Scholar

[21] Li, X.—Qaisar, S.—Nasir, J.—Butt, S. I.—Ahmad F.—Bari, M.—Farooq, S. E.: Some results on integral inequalities via Riemann–Liouville fractional integrals, J. Inequal. Appl. 2019 (2019), Art. No. 214.10.1186/s13660-019-2160-1Search in Google Scholar

[22] Liu, B. Z.: An inequality of Simpson type, Proc. Math. Phys. Eng. Sci. 461 (2005), 2155–2158.10.1098/rspa.2005.1505Search in Google Scholar

[23] Liu, W.: Some Simpson type inequalities for h-convex and (a,m)-convex functions, J. Comput. Anal. Appl. 16(5) (2014), 1005–1012.Search in Google Scholar

[24] Luo, C.—Du, T.: Generalized Simpson type inequalities involving Riemann-Liouville fractional integrals and their applications, Filomat 34(3) (2020), 751–760.10.2298/FIL2003751LSearch in Google Scholar

[25] Miller, S.—Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley, 1993.Search in Google Scholar

[26] Özdemir, M. E.—Akdemir, A. O.—Kavurmaci, H.: On the Simpson’s inequality for convex functions on the coordinates, Turkish Journal of Analysis and Number Theory 2(5) (2014), 165–169.10.12691/tjant-2-5-2Search in Google Scholar

[27] Özdemir, M. E.—Yildiz, C.: New inequalities for Hermite-Hadamard and Simpson type with applications, Tamkang J. Math. 44(2) (2013), 209–216.10.5556/j.tkjm.44.2013.1179Search in Google Scholar

[28] Park, J.: On Simpson-like type integral inequalities for differentiable preinvex functions, Appl. Math. Sci. 7(121) (2013), 6009–6021.10.12988/ams.2013.39498Search in Google Scholar

[29] Park, J.: On some integral inequalities for twice differentiable quasi-convex and convex functions via fractional integrals, Appl. Math. Sci. 9(62) (2015), 3057–3069.10.12988/ams.2015.53248Search in Google Scholar

[30] Rashid, S.—Akdemir, A. O.—Jarad, F.—Noor, M. A.—Noor, K. I.: Simpson’s type integral inequalities for k-fractional integrals and their applications, AIMS Math. 4(4) (2019), 1087–1100.10.3934/math.2019.4.1087Search in Google Scholar

[31] Sarikaya, M. Z.—Aktan, N.: On the generalization of some integral inequalities and their applications, Math. Comput. Modelling 54(9–10) (2011), 2175–2182.10.1016/j.mcm.2011.05.026Search in Google Scholar

[32] 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).Search in Google Scholar

[33] Sarikaya, M. Z.—Set, E.—Özdemir, M. E.: On new inequalities of Simpson’s type for s-convex functions, Comput. Math. Appl. 60(8) (2010), 2191–2199.10.1016/j.camwa.2010.07.033Search in Google Scholar

[34] Sarikaya, M. Z.—Set, E.—Özdemir, M. E.: On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex, J. Appl. Math. Stat. Inform. 9(1) (2013), 37–45.10.2478/jamsi-2013-0004Search in Google Scholar

[35] Sarikaya, M. Z.—Budak, H.—Erden, S.: On new inequalities of Simpson’s type for generalized convex functions, Korean J. Math. 27(2) (2019), 279–295.Search in Google Scholar

[36] Set, E.—Akdemir, A. O.—Özdemir, M. E.: Simpson type integral inequalities for convex functions via Riemann-Liouville integrals, Filomat 31(14) (2017), 4415–4420.10.2298/FIL1714415SSearch in Google Scholar

[37] Vivas-Cortez, M.—Abdeljawad, T.—Mohammed, P. M.—Rangel-Oliveros, Y.: Simpson’s integral inequalities for twice differentiable convex functions, Math. Probl. Eng. 2020 (2020), Art. ID 1936461.10.1155/2020/1936461Search in Google Scholar

[38] You, X.—Hezenci, F.—Budak, H.—Kara, H.: New Simpson type inequalities for twice differentiable functions via generalized fractional integrals, AIMS Math. 7(3) (2021), 3959–3971.10.3934/math.2022218Search in Google Scholar

Received: 2022-02-07
Accepted: 2022-06-16
Published Online: 2023-06-15

© 2023 Mathematical Institute Slovak Academy of Sciences

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