Research Article
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Year 2025, Volume: 13 Issue: 2, 212 - 222, 31.10.2025

Ayşe Nur Altunok , Tuba Tunç

Abstract

References

  • [1] T. Abdeljawad, S. Rashid, Z. Hammouch, ˙I.˙Is¸can and Y. M. Chu, Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications, Advances in Difference Equations, Vol:2020, No.1 (2020), 1-26.
  • [2] M. Alomari, M. Darus and S. S. Dragomir, New inequalities of Simpson’s type for s-convex functions with applications, RGMIA Res. Rep. Coll., Vol:12, No.4 (2009).
  • [3] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Science, Vol:20, No.2 (2016), 763-769.
  • [4] 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., Vol:44, No.17 (2021), 12522-12536.
  • [5] H. Budak, New version of Simpson type inequality for $\Psi$-Hilferfractional integrals, Adv. Anal. Appl. Math.,1(1) (2024), 1-11.
  • [6] H. Budak, F. Hezenci, H. Kara and M.Z. Sarikaya, Bounds for the error in approximating a fractional integral by Simpson’s Rule, Mathematics, Vol:11, No.10 (2023), 2282.
  • [7] H. Budak, F. Hezenci, T. Tunc¸ and M. Z. Sarikaya, On new versions of Hermite-Hadamard-type inequalities based on tempered fractional integrals, Filomat, Vol:38, No.7 (2024), 2361–2379.
  • [8] R. G. Buschman, Decomposition of an integral operator by use of Mikusinski calculus, SIAM J. Math. Anal., Vol:3, No.1 (1972), 83-85.
  • [9] J. Cai, B. Wang and T. Du, Simpson type inequalities for twice-differentiable functions arising from Tempered fractional integral operators, IAENG Int. J. Appl. Math., Vol:54, No.5 (2024), 831-839.
  • [10] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, Vol:1, No.2 (2015), 73-85.
  • [11] M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., Vol:55, No.1 (1994), 99-124.
  • [12] J. Chen and X. Huang, Some new inequalities of Simpson’s type for s-convex functions via fractional integrals, Filomat, Vol:31, No.15 (2017), 4989-4997.
  • [13] I. Demir and T. Tunc, A new approach to Simpson-type inequality with proportional Caputo-hybrid operator, Math. Methods Appl., Vol:48, No.1 (2025), 93-106.
  • [14] I. Demir and E. Unes¸, Conformable fractional Milne-type inequalities through twice-differentiable convex functions, Fundam. J. Math. Appl., Vol:8, No.1 (2025), 31-42.
  • [15] S. S. Dragomir, R. P. Agarwal and P. Cerone, On Simpson’s inequality and applications, J. Inequal. Appl. Vol:5, No.6 (2000), 533-579.
  • [16] 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., Vol:293, (2017), 358-369.
  • [17] F. Sabzikar, M.M. Meerschaert and J. Chen, Tempered fractional calculus, J. Comput. Phys., Vol:293, (2015), 14-28.
  • [18] Gorenflo, R. and Mainardi, F., Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien, 1997, 223-276.
  • [19] X. Hai and S.H. Wang, Simpson type inequalities for convex function based on the generalized fractional integrals, Turkish Journal of Inequalities, Vol:5, No.1 (2021), 1-15.
  • [20] F. Hezenci, H. Budak and H. Kara, New version of fractional Simpson type inequalities for twice differentiable functions, Advances in Difference Equations, Vol:2021, No.460 (2021).
  • [21] F. Hezenci, H. Budak, H. Kara and U. Bas¸, Novel results of Milne-type inequalities involving tempered fractional integrals, Bound. Value Probl., Vol:2024, No.1 (2024), 12.
  • [22] S. Hussain and S. Qaisar, More results on Simpson’s type inequality through convexity for twice differentiable continuous mappings, SpringerPlus, Vol:5, No.1 (2016), 1-9.
  • [23] S. Hussain, J. Khalid and Y. M. Chu, Some generalized fractional integral Simpson’s type inequalities with applications, AIMS Math, Vol:5, No.6 (2020), 5859-5883.
  • [24] S. Kermausuor, Simpson’s type inequalities via the Katugampola fractional integrals for s-convex functions, Kragujevac J. Math., Vol:45, No.5 (2021), 709-720.
  • [25] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
  • [26] H. Lei, G. Hu, J. Nie and T. Du, Generalized Simpson-type inequalities considering first derivatives through the k -Fractional Integrals, IAENG Int. J. Appl. Math., Vol:50, No.3 (2020), 1-8.
  • [27] C. Li, W. Deng and L. Zhao, Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discret. Cont. Dyn-B, Vol:24, No.4 (2019), 1989-2015.
  • [28] W. Liu, Some Simpson type inequalities for h-convex and (a;m)-convex functions, J. Comput. Analy. App., Vol:16, No.5 (2014), 1005-1012.
  • [29] C. Luo and T. Du, Generalized Simpson type inequalities involving Riemann-Liouville fractional integrals and their applications, Filomat, Vol:34, No.3 (2020), 751-760.
  • [30] M. Matloka, Some inequalities of Simpson type for h-convex functions via fractional integrals, Abstr. Appl. Anal., Vol:2015, Article ID 956850, 5 pages.
  • [31] F. Sabzikar, M. M. Meerschaert and J. Chen, Tempered fractional calculus, J. Comput. Phys., Vol:293, (2015), 14-28.
  • [32] P. O. Mohammed, M. Z. Sarikaya and D. Baleanu, On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry, Vol:12, No.4 (2020), 1-17.
  • [33] M. E. Ozdemir, A. O. Akdemir and H. Kavurmacı, On the Simpson’s inequality for convex functions on the coordinates, Turkish Journal of Analysis and Number Theory, Vol:2, No.5 (2014), 165-169.
  • [34] J. Park, On Some Integral Inequalities for twice differentiable quasi-convex and convex functions via fractional integrals, Appl. Math. Sci., Vol:9, No.62 (2015), 3057-3069.
  • [35] S. Rashid, A. O. Akdemir, F. Jarad, M. A. Noor and K. I. Noor, Simpson’s type integral inequalities for k-fractional integrals and their applications, AIMS Mathematics, Vol:4, No.4 (2019), 1087-1100.
  • [36] C. E. Torres Ledesma, H. A. Cuti, J. P. Avalos Rodr´ıguez and W. Zubiaga Vera, Some boundedness results for Riemann-Liouville tempered fractional integrals, Fractional Calculus and Applied Analysis, Vol: 27, No.2 (2024), 818-847.
  • [37] M. Z. Sarikaya, E. Set and M. E. O¨ zdemir, On new inequalities of Simpson’s type for s-convex functions, Comput. Math. Appl., Vol:60, No.8 (2020), 2191-2199.
  • [38] 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.
  • [39] X. You, F. Hezenci, H. Budak and H. Kara, New Simpson type inequalities for twice differentiable functions via generalized fractional integrals, AIMS Mathematics, Vol:7, No.3 (2021), 3959-3971.

Estimates on the Simpson Type Inequalities via Tempered Fractional Integrals

Year 2025, Volume: 13 Issue: 2, 212 - 222, 31.10.2025

Ayşe Nur Altunok , Tuba Tunç

Abstract

In this paper, we determine the upper and lower bounds for Simpson’s type inequalities involving tempered fractional integrals, focusing on functions with bounded second derivatives. Additionally, by considering specific parameter values in the results, we recover previous studies, thus generalizing the findings of earlier research.

Keywords

Simpson-type inequalities Riemann-Liouville fractional integrals tempered fractional integrals bounded second derivatives

References

  • [1] T. Abdeljawad, S. Rashid, Z. Hammouch, ˙I.˙Is¸can and Y. M. Chu, Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications, Advances in Difference Equations, Vol:2020, No.1 (2020), 1-26.
  • [2] M. Alomari, M. Darus and S. S. Dragomir, New inequalities of Simpson’s type for s-convex functions with applications, RGMIA Res. Rep. Coll., Vol:12, No.4 (2009).
  • [3] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Science, Vol:20, No.2 (2016), 763-769.
  • [4] 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., Vol:44, No.17 (2021), 12522-12536.
  • [5] H. Budak, New version of Simpson type inequality for $\Psi$-Hilferfractional integrals, Adv. Anal. Appl. Math.,1(1) (2024), 1-11.
  • [6] H. Budak, F. Hezenci, H. Kara and M.Z. Sarikaya, Bounds for the error in approximating a fractional integral by Simpson’s Rule, Mathematics, Vol:11, No.10 (2023), 2282.
  • [7] H. Budak, F. Hezenci, T. Tunc¸ and M. Z. Sarikaya, On new versions of Hermite-Hadamard-type inequalities based on tempered fractional integrals, Filomat, Vol:38, No.7 (2024), 2361–2379.
  • [8] R. G. Buschman, Decomposition of an integral operator by use of Mikusinski calculus, SIAM J. Math. Anal., Vol:3, No.1 (1972), 83-85.
  • [9] J. Cai, B. Wang and T. Du, Simpson type inequalities for twice-differentiable functions arising from Tempered fractional integral operators, IAENG Int. J. Appl. Math., Vol:54, No.5 (2024), 831-839.
  • [10] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, Vol:1, No.2 (2015), 73-85.
  • [11] M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., Vol:55, No.1 (1994), 99-124.
  • [12] J. Chen and X. Huang, Some new inequalities of Simpson’s type for s-convex functions via fractional integrals, Filomat, Vol:31, No.15 (2017), 4989-4997.
  • [13] I. Demir and T. Tunc, A new approach to Simpson-type inequality with proportional Caputo-hybrid operator, Math. Methods Appl., Vol:48, No.1 (2025), 93-106.
  • [14] I. Demir and E. Unes¸, Conformable fractional Milne-type inequalities through twice-differentiable convex functions, Fundam. J. Math. Appl., Vol:8, No.1 (2025), 31-42.
  • [15] S. S. Dragomir, R. P. Agarwal and P. Cerone, On Simpson’s inequality and applications, J. Inequal. Appl. Vol:5, No.6 (2000), 533-579.
  • [16] 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., Vol:293, (2017), 358-369.
  • [17] F. Sabzikar, M.M. Meerschaert and J. Chen, Tempered fractional calculus, J. Comput. Phys., Vol:293, (2015), 14-28.
  • [18] Gorenflo, R. and Mainardi, F., Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien, 1997, 223-276.
  • [19] X. Hai and S.H. Wang, Simpson type inequalities for convex function based on the generalized fractional integrals, Turkish Journal of Inequalities, Vol:5, No.1 (2021), 1-15.
  • [20] F. Hezenci, H. Budak and H. Kara, New version of fractional Simpson type inequalities for twice differentiable functions, Advances in Difference Equations, Vol:2021, No.460 (2021).
  • [21] F. Hezenci, H. Budak, H. Kara and U. Bas¸, Novel results of Milne-type inequalities involving tempered fractional integrals, Bound. Value Probl., Vol:2024, No.1 (2024), 12.
  • [22] S. Hussain and S. Qaisar, More results on Simpson’s type inequality through convexity for twice differentiable continuous mappings, SpringerPlus, Vol:5, No.1 (2016), 1-9.
  • [23] S. Hussain, J. Khalid and Y. M. Chu, Some generalized fractional integral Simpson’s type inequalities with applications, AIMS Math, Vol:5, No.6 (2020), 5859-5883.
  • [24] S. Kermausuor, Simpson’s type inequalities via the Katugampola fractional integrals for s-convex functions, Kragujevac J. Math., Vol:45, No.5 (2021), 709-720.
  • [25] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
  • [26] H. Lei, G. Hu, J. Nie and T. Du, Generalized Simpson-type inequalities considering first derivatives through the k -Fractional Integrals, IAENG Int. J. Appl. Math., Vol:50, No.3 (2020), 1-8.
  • [27] C. Li, W. Deng and L. Zhao, Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discret. Cont. Dyn-B, Vol:24, No.4 (2019), 1989-2015.
  • [28] W. Liu, Some Simpson type inequalities for h-convex and (a;m)-convex functions, J. Comput. Analy. App., Vol:16, No.5 (2014), 1005-1012.
  • [29] C. Luo and T. Du, Generalized Simpson type inequalities involving Riemann-Liouville fractional integrals and their applications, Filomat, Vol:34, No.3 (2020), 751-760.
  • [30] M. Matloka, Some inequalities of Simpson type for h-convex functions via fractional integrals, Abstr. Appl. Anal., Vol:2015, Article ID 956850, 5 pages.
  • [31] F. Sabzikar, M. M. Meerschaert and J. Chen, Tempered fractional calculus, J. Comput. Phys., Vol:293, (2015), 14-28.
  • [32] P. O. Mohammed, M. Z. Sarikaya and D. Baleanu, On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry, Vol:12, No.4 (2020), 1-17.
  • [33] M. E. Ozdemir, A. O. Akdemir and H. Kavurmacı, On the Simpson’s inequality for convex functions on the coordinates, Turkish Journal of Analysis and Number Theory, Vol:2, No.5 (2014), 165-169.
  • [34] J. Park, On Some Integral Inequalities for twice differentiable quasi-convex and convex functions via fractional integrals, Appl. Math. Sci., Vol:9, No.62 (2015), 3057-3069.
  • [35] S. Rashid, A. O. Akdemir, F. Jarad, M. A. Noor and K. I. Noor, Simpson’s type integral inequalities for k-fractional integrals and their applications, AIMS Mathematics, Vol:4, No.4 (2019), 1087-1100.
  • [36] C. E. Torres Ledesma, H. A. Cuti, J. P. Avalos Rodr´ıguez and W. Zubiaga Vera, Some boundedness results for Riemann-Liouville tempered fractional integrals, Fractional Calculus and Applied Analysis, Vol: 27, No.2 (2024), 818-847.
  • [37] M. Z. Sarikaya, E. Set and M. E. O¨ zdemir, On new inequalities of Simpson’s type for s-convex functions, Comput. Math. Appl., Vol:60, No.8 (2020), 2191-2199.
  • [38] 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.
  • [39] X. You, F. Hezenci, H. Budak and H. Kara, New Simpson type inequalities for twice differentiable functions via generalized fractional integrals, AIMS Mathematics, Vol:7, No.3 (2021), 3959-3971.
There are 39 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Research Article
Authors

Ayşe Nur Altunok 0009-0002-6116-583X

Tuba Tunç

Submission Date March 24, 2025
Acceptance Date May 9, 2025
Publication Date October 31, 2025
Published in Issue Year 2025 Volume: 13 Issue: 2

Cite

APA Altunok, A. N., & Tunç, T. (2025). Estimates on the Simpson Type Inequalities via Tempered Fractional Integrals. Konuralp Journal of Mathematics, 13(2), 212-222.
AMA Altunok AN, Tunç T. Estimates on the Simpson Type Inequalities via Tempered Fractional Integrals. Konuralp J. Math. October 2025;13(2):212-222.
Chicago Altunok, Ayşe Nur, and Tuba Tunç. “Estimates on the Simpson Type Inequalities via Tempered Fractional Integrals”. Konuralp Journal of Mathematics 13, no. 2 (October 2025): 212-22.
EndNote Altunok AN, Tunç T (October 1, 2025) Estimates on the Simpson Type Inequalities via Tempered Fractional Integrals. Konuralp Journal of Mathematics 13 2 212–222.
IEEE A. N. Altunok and T. Tunç, “Estimates on the Simpson Type Inequalities via Tempered Fractional Integrals”, Konuralp J. Math., vol. 13, no. 2, pp. 212–222, 2025.
ISNAD Altunok, Ayşe Nur - Tunç, Tuba. “Estimates on the Simpson Type Inequalities via Tempered Fractional Integrals”. Konuralp Journal of Mathematics 13/2 (October2025), 212-222.
JAMA Altunok AN, Tunç T. Estimates on the Simpson Type Inequalities via Tempered Fractional Integrals. Konuralp J. Math. 2025;13:212–222.
MLA Altunok, Ayşe Nur and Tuba Tunç. “Estimates on the Simpson Type Inequalities via Tempered Fractional Integrals”. Konuralp Journal of Mathematics, vol. 13, no. 2, 2025, pp. 212-2.
Vancouver Altunok AN, Tunç T. Estimates on the Simpson Type Inequalities via Tempered Fractional Integrals. Konuralp J. Math. 2025;13(2):212-2.
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