Generalizations of some fractional integral inequalities for m-convex functions via generalized Mittag-Leffler function
DOI:
https://doi.org/10.24193/subbmath.2018.1.02Keywords:
m-convex function, Hadamard inequality, generalized Mittag-Leffler function.Abstract
In this paper we are interested to present some general fractional integral inequalities for m-convex functions by involving generalized Mittag-Leffler function. In particular we produce inequalities for several kinds of fractional integrals. Also these inequalities have some connections with known integral inequalities.
Mathematics Subject Classification (2010): 26A51, 26A33, 33E12.
References
Dalir, M., Bashour, M., Applications of fractional calculus, Appl. Math. Sci., 4(2010), no. 21, 1021-1032.
Dragomir, S.S., On some new inequalities of Hermite-Hadamard type for m-convex functions, Turkish J. Math., 33(2002), no. 1, 45-55.
Dragomir, S.S., Agarwal, R.P., Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11(1998), no. 5, 91-95.
Dragomir, S.S., Toader, G.H., some inequalities for m-convex functions, Stud. Univ. Babe¸s-Bolyia. Math., 38(1993), no. 1, 21-28.
Farid, G., A treatment of the Hadamard inequality due to m-convexity via generalized fractional integrals, J. Fract. Calc. Appl. , 9(2018), no. 1, 8-14.
Farid, G., Hadamard and Fej´er-Hadamard inequalities for generalized fractional integrals involving special functions, Konuralp J. Math., 4(2016), no. 1, 108-113.
Farid, G., Marwan, M., Rehman, A.U., New mean value theorems and generalization of Hadamard inequality via coordinated m-convex functions, J. Inequal. Appl., Article ID 283, (2015), 11 pp.
Farid, G., Peˇcari´c, J., Tomovski, Z., Opial-type inequalities for fractional integral operator involving Mittag-Leffler function, Fract. Differ. Calc., 5(2015), no. 1, 93-106.
Farid, G., Rehman, A.U., Generalizations of some integral inequalities for fractional integrals, Ann. Math. Sil., to appear, https://doi.org/10.1515/amsil-2017-0010.
Iscan, I., New estimates on generalization of some integral inequalities for (α, m)-convex functions, Contemp. Anal. Appl. Math., 1(2013), no. 2, 253-264.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier, New York-London, 2006.
Laurent, H., Sur le calcul des derivees a indices quelconques, Nouv. Annales de Mathematiques, 3(1884), no. 3, 240-252.
Lazarevi´c, M., Advanced topics on applications of fractional calculus on control problems, System Stability and Modeling, WSEAS Press, 2014.
Letnikov, A.V., Theory of differentiation with an arbitray index (Russian), Moscow, Matem. Sbornik, 3, 1-66, 1868.
Miller, K., Ross, B., An introduction to the fractional differential equations, John, Wiley and Sons Inc., New York, 1993.
Mocanu, P.T., S¸erb, I., Toader, G., Real star convex functions, Stud. Univ. Babe¸s-Bolyai Math., 42(1997), no. 3, 65-80.
Prabhakar, T.R., A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19(1971), 7-15.
Salim, L.T.O., Faraj, A.W., A Generalization of Mittag-Leffler function and integral operator associated with integral calculus, J. Frac. Calc. Appl., 3(2012), no. 5, 1-13.
Sarikaya, M.Z., Erden, S., On the Hermite-Hadamard Fej´er type integral inequality for convex functions, Turkish Journal of Analysis and Number Theory, 2(2014), no. 3, 85-89.
Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N., Hermite-Hadamard inequalities for frac- tional integrals and related fractional inequalities, J. Math. Comput. Model, 57(2013), 2403-2407.
Sonin, N.Y., On differentiation with arbitray index, Moscow Matem. Sbornik, 6(1869), no. 1, 1-38.
Srivastava, H.M., Tomovski, Z., Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernal, Appl. Math. Comput., 211(2009), no. 1, 198-210.
Toader, G., Some generaliztion of convexity, Proc. Colloq. Approx. Optim, Cluj Napoca (Romania), (1984), 329-338.
Tomovski, Z., Hiller, R., Srivastava, H.M., Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler function, Integral Transforms Spec. Funct., 21(2011), no. 11, 797-814.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2018 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.