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On corrected Simpson-type inequalities via local fractional integrals

  • Abdelghani Lakhdari EMAIL logo , Badreddine Meftah and Wedad Saleh
Published/Copyright: June 26, 2024
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Abstract

The paper discusses corrected Simpson-type inequalities on fractal sets. Based on an introduced identity, we establish some error bounds for the considered formula using the generalized s-convexity and s-concavity of the local fractional derivative. Finally, we present some graphical representations justifying the established theoretical framework as well as some applications.

MSC 2020: 26D10; 26D15; 26A51

References

[1] T. Abdeljawad, S. Rashid, Z. Hammouch and Y.-M. Chu, Some new local fractional inequalities associated with generalized ( s , m ) -convex functions and applications, Adv. Difference Equ. 2020 (2020), Paper No. 406. 10.1186/s13662-020-02865-wSearch in Google Scholar

[2] G.-S. Chen, Generalizations of Hölder’s and some related integral inequalities on fractal space, J. Funct. Spaces Appl. 2013 (2013), Article ID 198405. 10.1155/2013/198405Search in Google Scholar

[3] T. Du, H. Wang, M. A. Khan and Y. Zhang, Certain integral inequalities considering generalized m-convexity on fractal sets and their applications, Fractals 27 (2019), no. 7, Article ID 1950117. 10.1142/S0218348X19501172Search in Google Scholar

[4] G. A. Edgar, Integral, Probability, and Fractal Measures, Springer, New York, 1998. 10.1007/978-1-4757-2958-0Search in Google Scholar

[5] Z. A. Khan, S. Rashid, R. Ashraf, D. Baleanu and Y.-M. Chu, Generalized trapezium-type inequalities in the settings of fractal sets for functions having generalized convexity property, Adv. Difference Equ. 2020 (2020), Paper No. 657. 10.1186/s13662-020-03121-xSearch in Google Scholar

[6] K. M. Kolwankar and A. D. Gangal, Local fractional calculus: A calculus for fractal space-time, Fractals: Theory and Applications in Engineering, Springer, London (1999), 171–181. 10.1007/978-1-4471-0873-3_12Search in Google Scholar

[7] A. Lakhdari, W. Saleh, B. Meftah and A. Iqbal, Corrected dual-Simpson-type inequalities for differentiable generalized convex functions on fractal set, Fractal Fract. 6 (2022), no. 12, Paper No. 710. 10.3390/fractalfract6120710Search in Google Scholar

[8] C. Luo, H. Wang and T. Du, Fejér–Hermite–Hadamard type inequalities involving generalized h-convexity on fractal sets and their applications, Chaos Solitons Fractals 131 (2020), Article ID 109547. 10.1016/j.chaos.2019.109547Search in Google Scholar

[9] B. Meftah, A. Lakhdari, W. Saleh and A. Kiliçman, Some new fractal Milne type integral inequalities via generalized convexity with applications, Fractal Fract. 7 (2023), no. 2, Paper No. 166. 10.3390/fractalfract7020166Search in Google Scholar

[10] B. Meftah, A. Souahi and M. Merad, Some local fractional Maclaurin type inequalities for generalized convex functions and their applications, Chaos Solitons Fractals 162 (2022), Article ID 112504. 10.1016/j.chaos.2022.112504Search in Google Scholar

[11] H. Mo and X. Sui, Generalized s-convex functions on fractal sets, Abstr. Appl. Anal. 2014 (2014), Article ID 254737. 10.1155/2014/636751Search in Google Scholar

[12] H. Mo and X. Sui, Hermite–Hadamard-type inequalities for generalized s-convex functions on real linear fractal set α ( 0 < α < 1 ), Math. Sci. (Springer) 11 (2017), no. 3, 241–246. 10.1007/s40096-017-0227-zSearch in Google Scholar

[13] J. E. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Math. Sci. Eng. 187, Academic Press, Boston, 1992. Search in Google Scholar

[14] W. Saleh, A. Lakhdari, O. Almutairi and A. Kiliçman, Some remarks on local fractional integral inequalities involving Mittag-Leffler kernel using generalized ( E , h ) -convexity, Mathematics 11 (2023), no. 6, Paper No. 1373. 10.3390/math11061373Search in Google Scholar

[15] W. Saleh, B. Meftah, A. Lakhdari and A. Kiliçman, Exploring the companion of Ostrowski’s inequalities via local fractional integrals, Eur. J. Pure Appl. Math. 16 (2023), no. 3, 1359–1380. 10.29020/nybg.ejpam.v16i3.4850Search in Google Scholar

[16] M. Z. Sarikaya, H. Budak and S. Erden, On new inequalities of Simpson’s type for generalized convex functions, Korean J. Math. 27 (2019), no. 2, 279–295. Search in Google Scholar

[17] M. Z. Sarikaya, T. Tunc and H. Budak, On generalized some integral inequalities for local fractional integrals, Appl. Math. Comput. 276 (2016), 316–323. 10.1016/j.amc.2015.11.096Search in Google Scholar

[18] W. Sun, Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, Fractals 27 (2019), no. 5, Article ID 1950071. 10.1142/S0218348X19500713Search in Google Scholar

[19] A.-M. Yang, Z.-S. Chen, H. M. Srivastava and X.-J. Yang, Application of the local fractional series expansion method and the variational iteration method to the Helmholtz equation involving local fractional derivative operators, Abstr. Appl. Anal. 2013 (2013), Article ID 259125. 10.1155/2013/259125Search in Google Scholar

[20] X.-J. Yang, Advanced Local Fractional Calculus and its Applications, World Science, New York, 2012. Search in Google Scholar

[21] Y.-J. Yang, D. Baleanu and X.-J. Yang, Analysis of fractal wave equations by local fractional Fourier series method, Adv. Math. Phys. 2013 (2013), Article ID 632309. 10.1155/2013/632309Search in Google Scholar

[22] Y. Yu and T. Du, Certain error bounds on the Bullen type integral inequalities in the framework of fractal space, J. Nonlinear Funct. Anal. 2022 (2022), Article ID 24. 10.23952/jnfa.2022.24Search in Google Scholar

[23] Y. Yu, J. Liu and T. Du, Certain error bounds on the parameterized integral inequalities in the sense of fractal sets, Chaos Solitons Fractals 161 (2022), Paper No. 112328. 10.1016/j.chaos.2022.112328Search in Google Scholar

Received: 2023-12-06
Accepted: 2024-02-27
Published Online: 2024-06-26
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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