On New Generalized Fractional Integral Operators and Related Fractional Inequalities

Yıl 2020, Cilt: 8 Sayı: 2, 268 - 278, 27.10.2020

Tuba Tunç , Hüseyin Budak , Fuat Usta , Mehmet Zeki Sarıkaya

Öz

In this paper, we define the generalized $k$-fractional integrals of a function with respect to the another function which generalizes many different types of fractional integrals such as Riemann-Liouville fractional, Hadamard fractional integrals, Katugampola fractional integral, $(k,s)$-fractional integral operators. Moreover, we obtain Hermite-Hadamard inequalities utilizing $k$-fractional integrals of a function with respect to the another function. We also investigate trapezoid inequalities for the functions whose derivatives in absolute value are convex. Finally, some special cases of these inequalities are given.

Anahtar Kelimeler

Fractional integral operators, Hermite-Hadamard inequality, midpoint inequality, convex function.

Kaynakça

• [1] R. P. Agarwal, M.-J. Luo and R. K. Raina, On Ostrowski type inequalities, Fasciculi Mathematici, 204, De Gruyter, doi:10.1515/fascmath-2016-0001, 2016.
• [2] A. Akkurt, M. E. Yıldırım, and H. Yıldırım, On some integral inequalities for (k;h)-Riemann-Liouville fractional integral, New Trends in Mathematical Science, 4 (2016), no. 2, 138–138.
• [3] N. Alp, M.Z.Sarikaya, M. Kunt and İ. İşcan, q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, Journal of King Saud University –Science, 30(2)(2018),193-203.
• [4] H. Budak, F. Usta, M. Z. Sarikaya and M. E. Özdemir, On generalization of midpoint type inequalities with generalized fractional integral operators, RACSAM, 113 (2019), 769-790.
• [5] H. Budak, F. Usta and M. Z. Sarikaya, Refinements of the Hermite–Hadamard iquality for co-ordinated convex mappings, Journal of Applied Analysis, 25 (2019), 73-81.
• [6] R. Diaz and E. Pariguan, On hypergeometric functions and Pochhammer k symbol, Divulg.Math, 15 (2007), 179-192. [7] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
• [8] F. Ertuğral, M.Z. Sarikaya and H. Budak, On Refinements of Hermite-Hadamard-Fejer type inequalities for fractional integral operators, Applications and Applied Mathematics: An International Journal, 13(1) (2018), 426-442.
• [9] M. Jleli and B. Samet, On Hermite-Hadamard type inequalities via fractional integrals of a function with respect to another function, J. Nonlinear Sci. Appl. 9 (2016), 1252-1260.
• [10] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218, 860-865, (2011).
• [11] A. A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Diferential Equations, Elsevier B.V., Amsterdam, Netherlands, 2006.
• [12] S. Mubeen and G. M. Habibullah, k-Fractional integrals and applications, International Journal of Contemporary Mathematical Sciences, 7(2012), 89–94.
• [13] J.E. Pecaric, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992.
• [14] R.K. Raina, On generalized Wright’s hypergeometric functions and fractional calculus operators, East Asian Math. J., 21(2) (2005), 191-203.
• [15] M. Z. Sarikaya, Z. Dahmani, M. E. Kiris and F. Ahmad, (k; s)-Riemann-Liouville fractional integral and applications, Hacettepe Journal of Mathematics and Statistics, 45(1) (2016), 77–89.
• [16] M.Z. Sarikaya, E. Set, H. Yaldiz and N., Basak, Hermite -Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, DOI:10.1016/j.mcm.2011.12.048, 57 (2013) 2403–2407.
• [17] M. Z. Sarikaya, H. Budak and F. Usta, Some generalized integral inequalities via fractional integrals, Acta Math. Univ. Comenianae, LXXXIX(1) (2020), 27-38.
• [18] T. Tunc¸ and M. Z. Sarikaya, On Hermite-Hadamard type inequalities via fractional integral operators, Filomat, 33(3), 837-854.
• [19] F. Usta, H. Budak, M. Z. Sarikaya and E. Set, On generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators, Filomat, 32(6) (2018), 2153-2171.
• [20] H. Yaldiz and M. Z. Sarikaya, On Hermite-Hadamard type inequalities for fractional integral operators, ResearchGate Article, Available online at: https://www.researchgate.net/publication/309824275.