A study on Milne-type inequalities for a specific fractional integral operator with applications
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Abstract
Fractional integral operators have been studied extensively in the last few decades by various mathematicians, because it plays a vital role in the developments of new inequalities. The main goal of the current study is to establish some new Milne-type inequalities by using the special type of fractional integral operator i.e Caputo Fabrizio operator. Additionally, generalization of these developed Milne-type inequalities for $s$-convex function are also given. Furthermore, applications to some special means, quadrature formula, and $q$-digamma functions are presented.
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References
[1] H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213–231. Google Scholar
[2] V. V. Kulish, J. L. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Eng., 124 (2002) 803–806. https://doi.org/10.1115/1.1478062 Google Scholar
[3] M. A. El-Shaed, Fractional calculus model of the semilunar heart valve vibrations, International design engineering technical conferences and computers and information in engineering confer-ence., (2003). Google Scholar
[4] M. W. Alomari, A companion of the generalized trapezoid inequality and applications, J. Math. Appl., 36 (2013) 5–15. Google Scholar
[5] S. S. Dragomir, On trapezoid quadrature formula and applications, Kragujev. J. Math., 23 (2001), 25–36. Google Scholar
[6] M. Z. Sarikaya, N. Aktan, On the generalization of some integral inequalities and their applica-tions, Math. Comput. Model., 54 (2011), 2175–2182. https://doi.org/10.1016/j.mcm.2011.05.026 Google Scholar
[7] M. Z. Sarikaya, H. Budak, Some Hermite-Hadamard type integral inequalities for twice differen-tiable mappings via fractional integrals, Facta Univ., Ser. Math. Inform. 29 (4) (2014), 371–384. Google Scholar
[8] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput. 147 (5) (2004), 137–146. https://doi.org/10.1016/S0096-3003(02)00657-4 Google Scholar
[9] S. S. Dragomir, On the midpoint quadrature formula for mappings with bounded variation and applications, Kragujev. J. Math., 22, 13 (2000) 19. http://eudml.org/doc/253402 Google Scholar
[10] M. Z. Sarikaya, A. Saglam, H. Yaldiz, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are convex and quasi-convex, Int. J. open probl. comput. sci. math. 5 (3) (2012). https://doi.org/10.48550/arXiv.1005.0451 Google Scholar
[11] S. S. Dragomir, R. P. Agarwal, P. Cerone, On Simpson’s inequality and applications, J. inequal. appl. 5 (2000), 533–579. Google Scholar
[12] M. Z. Sarikaya, E. Set, M. E. O ̈zdemir, On new inequalities of Simpson’s type for s-convex functions, Comput. math. appl. 60 (8) (2000), 2191–2199. https://doi.org/10.1016/j.camwa.2010.07.033 Google Scholar
[13] T. Du, Y. Li, Z. Yang, A generalization of Simpson’s inequality via differentiable mapping using extended (s, m)-convex functions, Appl. math. comput. 293 (2017), 358–369. https://doi.org/10.1016/j.amc.2016.08.045 Google Scholar
[14] S. S. Dragomir, On Simpson’s quadrature formula for mappings of bounded variation and appli-cations, Tamkang J. math., 30 (1) (1999), 53–58. Google Scholar
[15] H. Yang, S. Qaisar, A. Munir, and M. Naeem, New inequalities via Caputo-Fabrizio integral operator with applications, Aims Mathe, 8 (8) (2023), 19391–19412. https://doi.org/10.3934/math.2023989 Google Scholar
[16] N. A. Alqahtani., S. Qaisar, A. Munir, M. Naeem, & H. Budak, Error bounds for fractional integral inequalities with applications, Fractal and Fractional 8 (4), 208, (2024). https://doi.org/10.3390/fractalfract8040208 Google Scholar
[17] M. U. D. Junjua, A. Qayyum, A. Munir, H. Budak, M. M. Saleem, & S. S. A. Supadi, A study of some new Hermite–Hadamard inequalities via specific convex functions with applications, Mathematics 12 (3) (2024), 478. https://doi.org/10.3390/math12030478 Google Scholar
[18] S. K. Paul, L. N. Mishra, V. N. Mishra., & D. Baleanu, Analysis of mixed type nonlinear Volterra-Fredholm integral equations involving the Erd ́elyi-Kober fractional operator, Journal of king saud university-science 35 (10), (2023), 102949. https://doi.org/10.1016/j.jksus.2023.102949 Google Scholar
[19] V. K. Pathak, & L. N. Mishra, On solvbility and approximatting the soultions for nonlinear infinite system of frational functions integral equations in sequence space lp, p > 1, Journal of integral equations and applications 35 (4) (2023), 443–458. Google Scholar
[20] S. K. Paul, & L. N. Mishra, Stability analysis through the Bielecki metric to nonlinear fractional integral equations of n-product operators, Aims Mathe. 9 (4) (2024), 7770–7790. http://dx.doi.org/10.3934/math.2024377 Google Scholar
[21] V. K. Pathak, & L. N. Mishra, & V. N. Mishra, On the solvability of a class of nonlinear functional integral equations involving Erd ́elyi–Kober fractional operator, Mathematical methods in the applied sciences 46 (13) (2023), 14340–14352. https://doi.org/10.1002/mma.9322 Google Scholar
[22] S. K. Paul, & L. N. Mishra, & D. Baleanu, An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator, AIMS Mathematics 8 (2023). https://doi.org/10.3934/math.2023891 Google Scholar
[23] M. Raiz, R.S. Rajawat, L. N. Mishra, , V. N. Mishra, Approximation on bivariate of Durrmeyer operators based on beta function, The Journal of Analysis. (2023). https://doi.org/10.1007/s41478-023-00639-7 Google Scholar
[24] M. Z. Sarikaya, E. Set, M. E. Ozdemir, On new inequalities of simpson’s type for s-convex functions, Comput. math. appl.,60 (2010), 2191–2199. https://doi.org/10.1016/j.camwa.2010.07.033 Google Scholar
[25] A. D. Booth. Numerical Methods, 3rd edn. Butterworths, California (1966). Google Scholar
[26] J. L. W. V. Jensen. Surles fonctions convexes et les inegalites entre les valeurs moyennes, Acta math. 30 (1905), 175–193. Google Scholar
[27] H. Hudzik, L. Maligranda, Some remarks on s-convex functions, Aequationes Math. 48 (1994), 100–111. Google Scholar
[28] S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993). Google Scholar
[29] H. Budak, P. K ̈osem, H. Kara, On new Milne-type inequalities for fractional integrals, J inequal appl., 10(2023). https://doi.org/10.1186/s13660-023-02921-5 Google Scholar
[30] M. Caputo, & M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in fractional differentiation & applications.1 (2) (2015), 73–85. Google Scholar
[31] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge unversity press, (1955). Google Scholar