Document Type : Research Paper
Authors
Department of Mathematics, Faculty of Science and Arts, Duzce University, Duzce, Turkey
Abstract
In this paper, we establish some Trapezoid and Midpoint type inequalities for generalized fractional integrals by utilizing the functions whose second derivatives are bounded . We also give some new inequalities for $k$-Riemann-Liouville fractional integrals as special cases of our main results. We also obtain some Hermite-Hadamard type inequalities by using the condition $f^{\prime }(a+b-x)\geq f^{\prime }(x)$ for all $x\in \left[ a,\frac{a+b}{2}\right] $ instead of convexity.
Keywords
[1] M.U. Awan, M.A. Noor, T.S. Du and K.I. Noor, New refinements of fractional Hermite-Hadamard inequality, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 113(1), (2019), pp. 21-29.
[2] H. Budak, M.Z. Sarikaya and M.K. Yildiz, Hermite-Hadamard type inequalities for F-convex function involving fractional integrals, Filomat, 32(16),(2018), pp. 5509-5518.
[3] H. Budak, On refinements of Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral operators, Int. J. Optim. Control. Theor. Appl. IJOCTA, 9(1), (2019), pp. 41-48.
[4] H. Budak, On Fejer type inequalities for convex mappings utilizing fractional integrals of a function with respect to another function, Results Math., 74(1), (2019), 29.
[5] H. Budak, H. Kara, M.Z. Sarikaya and M.E. Kiris, New extensions of the Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals, Miskolc Math. Notes, 21(2), 2020.
[6] H. Budak, F. Ertugral and M.Z. Sarikaya, New generalization of Hermite-Hadamard type inequalities via generalized fractional integrals, An. Univ. Craiova Ser. Mat. Inform., 2020.
[7] F.X. Chen, Extensions of the Hermite-Hadamard inequality for convex functions via fractional integrals, J. Math. Inequal, (2016), 10(1), pp. 75-81.
[8] F.X. Chen, On the generalization of some Hermite-Hadamard Inequalities for functions with convex absolute values of the second derivatives via fractional integrals, Ukrainian Math. J., 12(70), (2019), pp. 1953-1965.
[9] S.S. Dragomir and C.E.M. Pearce, Selected topics on Hermite--Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000. Online: https://rgmia.org/papers/monographs/Master.pdf.
[10] S.S. Dragomir, Some inequalities of Hermite-Hadamard type for symmetrized convex functions and Riemann-Liouville fractional integrals, RGMIA Res. Rep. Coll., 20 (2017).
[11] S.S. Dragomir, P. Cerone and A. Sofo, Some remarks on the midpoint rule in numerical integration, Stud. Univ. Babe¸s-Bolyai Math., XLV(1), (2000), pp. 63-74.
[12] S.S. Dragomir, P. Cerone and A. Sofo, Some remarks on the trapezoid rule in numerical integration, Indian J. Pure Appl. Math., 31(5), (2000), pp. 475-494.
[13] A. Gozpinar, E. Set and S.S. Dragomir, Some generalized Hermite-Hadamard type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are s-convex, Acta Math. Univ. Comenian., 88(1), (2019), pp. 87-100.
[14] S.R. Hwang and K.L. Tseng, New Hermite-Hadamard-type inequalities for fractional integrals and their applications, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 112(4), (2018), pp. 1211-1223.
[15] 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(3), (2016), pp. 1252-1260.
[16] M.A. Khan, A. Iqbal, M. Suleman and Y.-M. Chu, Hermite-Hadamard type inequalities for fractional integrals via Green's function, J. Inequal. Appl., 2018 (2018), Article ID 161.
[17] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
[18] K. Liu, J. Wang and D. O'Regan, On the Hermite--Hadamard type inequality for $psi$-Riemann-Liouville fractional integrals via convex functions, J. Inequal. Appl., 2019 (2019), Article ID 27.
[19] S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993.
[20] P.O. Mohammed and M.Z. Sarikaya, Hermite-Hadamard type inequalities for F-convex function involving fractional integrals, J. Inequal. Appl., 2018 (2018), Article ID 359.
[21] S. Mubeen and G.M. Habibullah, $k$ -Fractional integrals and application, Int. J. Contemp. Math. Sciences, 7(2), (2012), pp. 89-94.
[22] N. Minculete and F-C. Mitroi, Fejer-type inequalities, Aust. J. Math. Anal. Appl., 9(1), (2012), Art. 12.
[23] J.E. Pecaric, F. Proschan and Y.L. Tong, Convex functions, partial orderings and statistical applications, Academic Press, Boston, 1992.
[24] S. Qaisar, M. Iqbal, S. Hussain, S. Butt and M.A. Meraj, New inequalities on Hermite-Hadamard utilizing fractional integrals, Kragujevac J. Math., 42(1), (2018), pp. 15-27.
[25] K. Qiu and J.R. Wang, A fractional integral identity and its application to fractional Hermite-Hadamard type inequalities, Journal of Interdisciplinary Mathematics, 21(1), (2018), pp. 1-16.
[26] M.Z. Sarikaya and N. Aktan, On the generalization some integral inequalities and their applications, Math. Comput. Model., 54 (2011), pp. 2175-2182.
[27] M.Z. Sarikaya and H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17(2), (2016), pp. 1049-1059.
[28] M.Z. Sarikaya and F. Ertugral, On the generalized Hermite-Hadamard inequalities, Annals of the University of Craiova-Mathematics and Computer Science Series, 47(1), (2020), pp. 193–213.
[29] M.Z. Sarikaya, On Fejer type inequalities via fractional integrals, J. Interdisciplinary Math., 21(1), (2018), pp. 143-155.
[30] M.Z. Sarikaya, E. Set, H. Yaldiz and N., Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), pp. 2403-2407.
[31] E. Set, A. Akdemir and B. Celik, On generalization of Fejer type inequalities via fractional integral operator, Filomat, 32(16), (2018), pp. 5537-5547.
[32] T. Tunc, S. Sonmezoglu and M.Z. Sarikaya, On integral inequalities of Hermite-Hadamard type via Green function and applications, Appl. Appl. Math., 14(1), (2019), pp. 452-462.
[2] H. Budak, M.Z. Sarikaya and M.K. Yildiz, Hermite-Hadamard type inequalities for F-convex function involving fractional integrals, Filomat, 32(16),(2018), pp. 5509-5518.
[3] H. Budak, On refinements of Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral operators, Int. J. Optim. Control. Theor. Appl. IJOCTA, 9(1), (2019), pp. 41-48.
[4] H. Budak, On Fejer type inequalities for convex mappings utilizing fractional integrals of a function with respect to another function, Results Math., 74(1), (2019), 29.
[5] H. Budak, H. Kara, M.Z. Sarikaya and M.E. Kiris, New extensions of the Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals, Miskolc Math. Notes, 21(2), 2020.
[6] H. Budak, F. Ertugral and M.Z. Sarikaya, New generalization of Hermite-Hadamard type inequalities via generalized fractional integrals, An. Univ. Craiova Ser. Mat. Inform., 2020.
[7] F.X. Chen, Extensions of the Hermite-Hadamard inequality for convex functions via fractional integrals, J. Math. Inequal, (2016), 10(1), pp. 75-81.
[8] F.X. Chen, On the generalization of some Hermite-Hadamard Inequalities for functions with convex absolute values of the second derivatives via fractional integrals, Ukrainian Math. J., 12(70), (2019), pp. 1953-1965.
[9] S.S. Dragomir and C.E.M. Pearce, Selected topics on Hermite--Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000. Online: https://rgmia.org/papers/monographs/Master.pdf.
[10] S.S. Dragomir, Some inequalities of Hermite-Hadamard type for symmetrized convex functions and Riemann-Liouville fractional integrals, RGMIA Res. Rep. Coll., 20 (2017).
[11] S.S. Dragomir, P. Cerone and A. Sofo, Some remarks on the midpoint rule in numerical integration, Stud. Univ. Babe¸s-Bolyai Math., XLV(1), (2000), pp. 63-74.
[12] S.S. Dragomir, P. Cerone and A. Sofo, Some remarks on the trapezoid rule in numerical integration, Indian J. Pure Appl. Math., 31(5), (2000), pp. 475-494.
[13] A. Gozpinar, E. Set and S.S. Dragomir, Some generalized Hermite-Hadamard type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are s-convex, Acta Math. Univ. Comenian., 88(1), (2019), pp. 87-100.
[14] S.R. Hwang and K.L. Tseng, New Hermite-Hadamard-type inequalities for fractional integrals and their applications, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 112(4), (2018), pp. 1211-1223.
[15] 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(3), (2016), pp. 1252-1260.
[16] M.A. Khan, A. Iqbal, M. Suleman and Y.-M. Chu, Hermite-Hadamard type inequalities for fractional integrals via Green's function, J. Inequal. Appl., 2018 (2018), Article ID 161.
[17] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
[18] K. Liu, J. Wang and D. O'Regan, On the Hermite--Hadamard type inequality for $psi$-Riemann-Liouville fractional integrals via convex functions, J. Inequal. Appl., 2019 (2019), Article ID 27.
[19] S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993.
[20] P.O. Mohammed and M.Z. Sarikaya, Hermite-Hadamard type inequalities for F-convex function involving fractional integrals, J. Inequal. Appl., 2018 (2018), Article ID 359.
[21] S. Mubeen and G.M. Habibullah, $k$ -Fractional integrals and application, Int. J. Contemp. Math. Sciences, 7(2), (2012), pp. 89-94.
[22] N. Minculete and F-C. Mitroi, Fejer-type inequalities, Aust. J. Math. Anal. Appl., 9(1), (2012), Art. 12.
[23] J.E. Pecaric, F. Proschan and Y.L. Tong, Convex functions, partial orderings and statistical applications, Academic Press, Boston, 1992.
[24] S. Qaisar, M. Iqbal, S. Hussain, S. Butt and M.A. Meraj, New inequalities on Hermite-Hadamard utilizing fractional integrals, Kragujevac J. Math., 42(1), (2018), pp. 15-27.
[25] K. Qiu and J.R. Wang, A fractional integral identity and its application to fractional Hermite-Hadamard type inequalities, Journal of Interdisciplinary Mathematics, 21(1), (2018), pp. 1-16.
[26] M.Z. Sarikaya and N. Aktan, On the generalization some integral inequalities and their applications, Math. Comput. Model., 54 (2011), pp. 2175-2182.
[27] M.Z. Sarikaya and H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17(2), (2016), pp. 1049-1059.
[28] M.Z. Sarikaya and F. Ertugral, On the generalized Hermite-Hadamard inequalities, Annals of the University of Craiova-Mathematics and Computer Science Series, 47(1), (2020), pp. 193–213.
[29] M.Z. Sarikaya, On Fejer type inequalities via fractional integrals, J. Interdisciplinary Math., 21(1), (2018), pp. 143-155.
[30] M.Z. Sarikaya, E. Set, H. Yaldiz and N., Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), pp. 2403-2407.
[31] E. Set, A. Akdemir and B. Celik, On generalization of Fejer type inequalities via fractional integral operator, Filomat, 32(16), (2018), pp. 5537-5547.
[32] T. Tunc, S. Sonmezoglu and M.Z. Sarikaya, On integral inequalities of Hermite-Hadamard type via Green function and applications, Appl. Appl. Math., 14(1), (2019), pp. 452-462.
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