Abstract
In this research article, an equality is established for Riemann–Liouville fractional integral. With the help of this equality, some corrected Euler–Maclaurin-type inequalities are established for the case of differentiable convex functions by using to the well-known Riemann–Liouville fractional integrals. Several important inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Moreover, we give example using graph in order to show that our main results are correct. Furthermore, we provide our results by using special cases of obtained theorems.
Similar content being viewed by others
Availability of data materials
Data sharing not applicable to this paper as no data sets were generated or analysed during the current study.
References
Agarwal, P., J. Tariboon, and S.K. Ntouyas. 2016. Some generalized Riemann–Liouville \(k\)-fractional integral inequalities. Journal of Inequalities and Applications 2016: 122.
Agarwal, P. 2017. Some inequalities involving Hadamard-type \(k\) -fractional integral operators. Mathematical Methods in the Applied Sciences 40 (11): 3882–3891.
Agarwal, P., M. Jleli, and M. Tomar. 2017. Certain Hermite–Hadamard type inequalities via generalized \(k\)-fractional integrals. Journal of Inequalities and Applications 2017: 55.
Budak, H., F. Hezenci, and H. Kara. 2021. On parametrized inequalities of Ostrowski and Simpson type for convex functions via generalized fractional integral. Mathematical Methods in the Applied Sciences 44 (30): 12522–12536.
Budak, H., F. Hezenci, and H. Kara. 2021. On generalized Ostrowski, Simpson and trapezoidal type inequalities for co-ordinated convex functions via generalized fractional integrals. Advances in Difference Equations 2021: 1–32.
Davis, P.J., and P. Rabinowitz. 1975. Methods of numerical integration. New York-San Francisco-London: Academic Press.
Lj, Dedić, M. Matić, and J. Pečarić. 2003. Euler–Maclaurin formulae. Mathematical Inequalities & Applications 6 (2): 247–275.
Dedić, L.J., M. Matić, J. Pečarić, and A. Vukelic. 2011. On Euler–Simpson 3/8 formulae. Nonlinear Studies 18 (1): 1–26.
Dragomir, S.S., R.P. Agarwal, and P. Cerone. 2000. On Simpson’s inequality and applications. Journal of Inequalities and Applications 5: 533–579.
Dragomir, S.S. 1999. On Simpson’s quadrature formula for mappings of bounded variation and applications. Tamkang Journal of Mathematics 30: 53–58.
Erden, S., S. Iftikhar, P. Kumam, and M.U. Awan. 2020. Some Newton’s like inequalities with applications. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 114 (4): 1–13.
Franjić, I., and J. Pečarić. 2005. Corrected Euler-Maclaurin’s formulae. Rendiconti del Circolo Matematico di Palermo 54: 259–272.
Franjić, I., and J. Pečarić. 2006. On corrected Euler–Simpson’s \(3/8\) formulae. Nonlinear Studies 13 (4): 309–319.
Gao, S., and W. Shi. 2012. On new inequalities of Newton’s type for functions whose second derivatives absolute values are convex. International Journal of Pure and Applied Mathematics 74 (1): 33–41.
Gorenflo, R., and F. Mainardi. 1997. Fractional calculus: Integral and differential equations of fractional order. Wien: Springer Verlag.
Hezenci, F., H. Budak, and H. Kara. 2021. New version of Fractional Simpson type inequalities for twice differentiable functions. Advances in Difference Equations 2021: 460.
Hezenci, F., H. Budak, P. Kosem. On New version of Newton’s inequalities for Riemann–Liouville fractional integrals. Rocky Mountain Journal of Mathematics accepted (in press).
Hezenci, F., H. Budak. Some Perturbed Newton type inequalities for Riemann–Liouville fractional integrals. Rocky Mountain Journal of Mathematics accepted (in press).
Iftikhar, S., P. Kumam, and S. Erden. 2020. Newton’s-type integral inequalities via local fractional integrals. Fractals 28 (03): 2050037.
Iftikhar, S., S. Erden, P. Kumam, and M.U. Awan. 2020. Local fractional Newton’s inequalities involving generalized harmonic convex functions. Advances in Difference Equations 2020 (1): 1–14.
Park, J. 2013. On Simpson-like type integral inequalities for differentiable preinvex functions. Applied Mathematical Sciences 7 (121): 6009–6021.
Kilbas, A.A., H.M. Srivastava, and J.J. Trujillo. 2006. Theory and applications of fractional differential equations. Amsterdam: Elsevier.
Noor, M.A., K.I. Noor, and S. Iftikhar. 2016. Some Newton’s type inequalities for harmonic convex functions. Journal of Advanced Mathematical Studies 9 (1): 07–16.
Pečarić, J.E., F. Proschan, Y.L. Tong. 1992. Convex functions. In Partial orderings and statistical applications. Boston: Academic Press.
Sitthiwirattham, T., K. Nonlaopon, M.A. Ali, and H. Budak. 2022. Riemann-Liouville fractional Newton’s type inequalities for differentiable convex functions. Fractal and Fractional 6 (3): 175.
You, X., F. Hezenci, H. Budak, and H. Kara. 2021. New Simpson type inequalities for twice differentiable functions via generalized fractional integrals. AIMS Mathematics 7 (3): 3959–3971.
Funding
There is no funding.
Author information
Authors and Affiliations
Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by S. Ponnusamy.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hezenci, F., Budak, H. Some Riemann–Liouville fractional integral inequalities of corrected Euler–Maclaurin-type. J Anal 32, 1309–1330 (2024). https://doi.org/10.1007/s41478-024-00753-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41478-024-00753-0