Citation: Hüseyin Budak, Ebru Pehlivan. Weighted Ostrowski, trapezoid and midpoint type inequalities for RiemannLiouville fractional integrals[J]. AIMS Mathematics, 2020, 5(3): 1960-1984. doi: 10.3934/math.2020131
[1] | R. P. Agarwal, M. J. Luo, R. K. Raina, On Ostrowski type inequalities, Fasciculi Math., 56 (2016), 5-27. doi: 10.1515/fascmath-2016-0001 |
[2] | M. Alomari, M. Darus, U. S. Kirmaci, Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means, Comput. Math. Appl., 59 (2010), 225-232. doi: 10.1016/j.camwa.2009.08.002 |
[3] | G. A. Anastassiou, General fractional Hermite-Hadamard inequalities using m-convexity and (s, m)-convexity, Front. Time Scales Inequal., 237 (2016), 255. |
[4] | A. G. Azpeitia, Convex functions and the Hadamard inequality, Rev. Colombiana Math., 28 (1994), 7-12. |
[5] | Y. Başcı, D. Baleanu, Ostrowski type Inequalities Involving ψ-Hilfer fractional Integrals, Mathematics, 7 (2019), 770. |
[6] | H. Budak, M. Z. Sarikaya, E. Set, Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense, J. Appl. Math. Comput. Mech., 15 (2016), 11-22. |
[7] | J. de la Cal, J. Carcamob, L. Escauriaza, A general multidimensional Hermite-Hadamard type inequality, J. Math. Anal. Appl., 356 (2009), 659-663. doi: 10.1016/j.jmaa.2009.03.044 |
[8] | H. Chen, U. N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274-1291. doi: 10.1016/j.jmaa.2016.09.018 |
[9] | S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000, 2004, Available from: http://rgmia.org/papers/monographs/Master2.pdf. |
[10] | S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. lett., 11 (1998), 91-95. |
[11] | S. S. Dragomir, Ostrowski type inequalities for Riemann-Liouville fractional integrals of bounded variation, Hölder and Lipschitzian functions, RGMIA Res. Rep. Coll., 20 (2017), 48. |
[12] | G. Farid, A. ur Rehman, M. Zahra, On Hadamard type inequalities for k-fractional integrals, Konurap J. Math., 4 (2016), 79-86. |
[13] | L. Fejer, Über die Fourierreihen, II. Math. Naturwiss. Anz Ungar. Akad. Wiss., 24 (1906), 369-390. |
[14] | R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Springer Verlag, Wien, 1997, 223-276. |
[15] | A. Guezane-Lakoud, F. Aissaoui, New fractional inequalities of Ostrowski type, Transylv. J. Math. Mech., 5 (2013), 103-106 |
[16] | M. Gúrbúz, Y. Taşdan, E. Set, Ostrowski type inequalities via the Katugampola fractional integrals, AIMS Mathematics, 5 (2020), 42-53. doi: 10.3934/math.2020004 |
[17] | M. Iqbal, S. Qaisar, M. Muddassar, A short note on integral inequality of type Hermite-Hadamard through convexity, J. Comput. Anal. Appl., 21 (2016), 946-953. |
[18] | I. Iscan, Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals, Stud. U. Babeş Bol. Math., 60 (2015), 355-366 |
[19] | İ. İşcan, On generalization of different type integral inequalities for s-convex functions via fractional integrals, Math. Sci. Appl., 2 (2014), 55-67. |
[20] | M. Jleli, 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. doi: 10.22436/jnsa.009.03.50 |
[21] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, Amsterdam, 2006. |
[22] | M. Kirane, B. T. Torebek, Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via fractional integrals, arXiv:1701.00092. |
[23] | U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 147 (2004), 137-146. |
[24] | W. Liu, A. Tuna, Diamond-α weighted Ostrowski type and Grüss type inequalities on time scales, Appl. Math. Comput., 270 (2015), 251-260. |
[25] | W. Liu, X. Gao, Y. Wen, Approximating the finite Hilbert transform via some companions of Ostrowski's inequalities, B. Malays. Math. Sci. So., 39 (2015), 1499-1513. |
[26] | W. Liu, A. Tuna, Y. Jiang, On weighted Ostrowski type, Trapezoid type, Grüss type and Ostrowski- Grüss like inequalities on time scales, Appl. Anal., 93 (2013), 551-571. |
[27] | S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993. |
[28] | M. A. Noor, M. U. Awan, Some integral inequalities for two kinds of convexities via fractional integrals, TJMM, 5 (2013), 129-136. |
[29] | M. A. Noor, K. I. Noor, M. U. Awan, Fractional Ostrowski inequalities for s-Godunova-Levin functions, Int. J. Anal. Appl., 5 (2014), 167-173. |
[30] | J. E. Pečarić, F. Proschan, Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992. |
[31] | A. M. Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 10 (1938), 226-227. |
[32] | M. E. Özdemir, M. Avcı-Ardıç, H. Kavurmacı-Önalan, Hermite-Hadamard type inequalities for s-convex and s-concave functions via fractional integrals, Turk. J. Sci., 1 (2016), 28-40. |
[33] | M. E. Ödemir, M. Avci, E. Set, On some inequalities of Hermite-Hadamard-type via m-convexity, Appl. Math. Lett., 23 (2010), 1065-1070. doi: 10.1016/j.aml.2010.04.037 |
[34] | M. E. Ödemir, M. Avci, H. Kavurmaci, Hermite-Hadamard-type inequalities via (α, m)-convexity, Comput. Math. Appl., 61 (2011), 2614-2620. doi: 10.1016/j.camwa.2011.02.053 |
[35] | I. Podlubni, Fractional Differential Equations, Academic Press, San Diego, 1999. |
[36] | S. Qaisar, S. Hussain, On Hermite-Hadamard type inequalities for functions whose first derivative absolute values are convex and concave, Fasciculi Math., 58 (2017), 155-166. doi: 10.1515/fascmath-2017-0011 |
[37] | A. Saglam, M. Z. Sarikaya, H. Yildirim, Some new inequalities of Hermite-Hadamard's type, Kyungpook Math. J., 50 (2010), 399-410. doi: 10.5666/KMJ.2010.50.3.399 |
[38] | M. Z. Sarikaya, H. Budak, Generalized Ostrowski type inequalities for local fractional integrals, P. Am. Math. Soc., 145 (2017), 1527-1538. |
[39] | M. Z. Sarikaya, H. Filiz, Note on the Ostrowski type inequalities for fractional integrals, Vietnam J. Math., 42 (2014), 187-190. doi: 10.1007/s10013-014-0056-4 |
[40] | M. Z. Sarikaya, E. Set, H. Yaldiz, et al. Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403-2407. doi: 10.1016/j.mcm.2011.12.048 |
[41] | M. Z. Sarikaya, H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 7 (2016), 1049-1059. |
[42] | M. Z. Sarikaya, H. Budak, Generalized Hermite-Hadamard type integral inequalities for fractional integrals, Filomat, 30 (2016), 1315-1326. doi: 10.2298/FIL1605315S |
[43] | M. Z. Sarikaya, A. Akkurt, H. Budak, et al. Hermite-hadamard's inequalities for conformable fractional integrals, RGMIA Res. Rep. Coll., 19 (2016), 58140. |
[44] | E. Set, M. E. Ozdemir, M. Z. Sarikaya, New inequalities of Ostrowski's type for s-convex functions in the second sense with applications, Facta U. Ser. Math. Inform., 27 (2012), 67-82. |
[45] | E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012), 1147-1154. doi: 10.1016/j.camwa.2011.12.023 |
[46] | E. Set, M. Z. Sarikaya, M. E. Ozdemir, et al. The Hermite-Hadamard's inequality for some convex functions via fractional integrals and related results, J. Appl. Math. Stat. Inform., 10 (2014), 69-83. doi: 10.2478/jamsi-2014-0014 |
[47] | E. Set,İ. İşcan, M. Z. Sarikaya, et al. On new inequalities of Hermite-Hadamard-Fejért ype for convex functions via fractional integrals, Appl. Math. Comput., 259 (2015), 875-881. |
[48] | E. Set, A. O. Akdemir, I. Mumcu, Ostrowski type inequalities for functions whoose derivatives are convex via conformable fractional integrals, J. Adv. Math. Stud., 10 (2017), 386-395. |
[49] | E. Set, A. O. Akdemir, A. Gözpınar, et al. Ostrowski type inequalities via new fractional conformable integrals, AIMS Mathematics, 4 (2019), 1684-1697. doi: 10.3934/math.2019.6.1684 |
[50] | S. F. Tahir, M. Mushtaq, M. Muddassar, A Note on integral inequalities on time scales associated with Ostrowski's type, J. Funct. Space., 2019 (2019), 4748373. |
[51] | K. L. Tseng, G. S. Yang, K. C. Hsu, Some inequalities for differentiable mappings and applications to Fejer inequality and weighted trapozidal formula, Taiwanese J. Math., 15 (2011), 1737-1747. doi: 10.11650/twjm/1500406376 |
[52] | J. Wang, X. Li, M. Fečkan, et al. Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal., 92 (2012), 2241-2253. |
[53] | J. Wang, X. Li, C. Zhu, Refinements of Hermite-Hadamard type inequalities involving fractional integrals, Bull. Belg. Math. Soc. Simon Stevin, 20 (2013), 655-666. doi: 10.36045/bbms/1382448186 |
[54] | H. Yildirim, Z. Kirtay, Ostrowski inequality for generalized fractional integral and related inequalities, Malaya J. Mat., 2 (2014), 322-329. |
[55] | C. Yildiz, M. E. Özdemir, M. Z. Sarikaya, New generalizations of Ostrowski-like type inequalities for fractional integrals, Kyungpook Math. J., 56 (2016), 161-172. doi: 10.5666/KMJ.2016.56.1.161 |
[56] | Y. Zhang, J. Wang, On some new Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals, J. Inequal. Appl., 2013 (2013), 220. |