Trapezoid type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation

Silvestru Sever Dragomir

doi:10.2478/ausm-2020-0003

Abstract

In this paper we establish some trapezoid type inequalities for the Riemann-Liouville fractional integrals of functions of bounded variation and of Hölder continuous functions. Applications for the g-mean of two numbers are provided as well. Some particular cases for Hadamard fractional integrals are also provided.

References

[1] A. Aglić Aljinović, Montgomery identity and Ostrowski type inequalities for Riemann-Liouville fractional integral, J. Math., 2014, Art. ID 503195, 6 pp.10.1155/2014/503195
Search in Google Scholar Back to article
[2] A. O. Akdemir, Inequalities of Ostrowski’s type for m- and (α, m)-logarithmically convex functions via Riemann-Liouville fractional integrals, J. Comput. Anal. Appl., 16 (2014), no. 2, 375–383
Search in Google Scholar Back to article
[3] G. A. Anastassiou, Fractional representation formulae under initial conditions and fractional Ostrowski type inequalities, Demonstr. Math., 48 (2015), no. 3, 357–37810.1515/dema-2015-0025
Search in Google Scholar Back to article
[4] G. A. Anastassiou, The reduction method in fractional calculus and fractional Ostrowski type inequalities, Indian J. Math., 56 (2014), no. 3, 333–357.
Search in Google Scholar Back to article
[5] 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), no. 4, 11–21.10.17512/jamcm.2016.4.02
Search in Google Scholar Back to article
[6] P. Cerone and S. S. Dragomir, Midpoint-type rules from an inequalities point of view, Handbook of analytic-computational methods in applied mathematics, 135–200, Chapman & Hall/CRC, Boca Raton, FL, 2000.10.1201/9780429123610-4
Search in Google Scholar Back to article
[7] S. S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comput. Math. Appl., 38 (1999), no. 11-12, 33–37.10.1016/S0898-1221(99)00282-5
Search in Google Scholar Back to article
[8] S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc.60 (1999), No. 3, 495–508.10.1017/S0004972700036662
Search in Google Scholar Back to article
[9] S. S. Dragomir, On the midpoint quadrature formula for mappings with bounded variation and applications, Kragujevac J. Math., 22 (2000), 13–19.
Search in Google Scholar Back to article
[10] S. S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Math. Ineq. Appl., 4 (2001), No. 1, 59-66. Preprint: RGMIA Res. Rep. Coll.2 (1999), Art. 7, [Online: http://rgmia.org/papers/v2n1/v2n1-7.pdf]10.7153/mia-04-05
Search in Google Scholar Back to article
[11] S. S. Dragomir, Refinements of the Ostrowski inequality in terms of the cumulative variation and applications, Analysis (Berlin) 34 (2014), No. 2, 223–240. Preprint: RGMIA Res. Rep. Coll., 16 (2013), Art. 29 [Online:http://rgmia.org/papers/v16/v16a29.pdf].10.1515/anly-2014-1252
Search in Google Scholar Back to article
[12] S. S. Dragomir, Ostrowski type inequalities for Lebesgue integral: a survey of recent results, Australian J. Math. Anal. Appl., Volume 14, Issue 1, Article 1, pp. 1-287, 2017. [Online http://ajmaa.org/cgi-bin/paper.pl?string=v14n1/V14I1P1.tex].
Search in Google Scholar Back to article
[13] S. S. Dragomir, Ostrowski type inequalities for Riemann-Liouville fractional integrals of bounded variation, Hölder and Lipschitzian functions, Preprint RGMIA Res. Rep. Coll., 20 (2017), Art. 48. [Online http://rgmia.org/papers/v20/v20a48.pdf].
Search in Google Scholar Back to article
[14] S. S. Dragomir, Ostrowski type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation, Preprint RGMIA Res. Rep. Coll., 20 (2017), Art 58. [Online http://rgmia.org/papers/v20/v20a58.pdf].
Search in Google Scholar Back to article
[15] S. S. Dragomir, M. S. Moslehian and Y. J. Cho, Some reverses of the Cauchy-Schwarz inequality for complex functions of self-adjoint operators in Hilbert spaces, Math. Inequal. Appl., 17 (2014), no. 4, 1365–1373.10.7153/mia-17-100
Search in Google Scholar Back to article
[16] A. Guezane-Lakoud and F. Aissaoui, New fractional inequalities of Ostrowski type, Transylv. J. Math. Mech., 5 (2013), no. 2, 103–106
Search in Google Scholar Back to article
[17] A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized (s, m, ϕ)-preinvex functions, Aust. J. Math. Anal. Appl., 13 (2016), no. 1, Art. 16, 11 pp.10.12988/nade.2016.6641
Search in Google Scholar Back to article
[18] A. Kilbas, A; H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. xvi+523 pp. ISBN: 978-0-444-51832-3; 0-444-51832-0
Search in Google Scholar Back to article
[19] M. A. Noor, K. I.Noor and S. Iftikhar, Fractional Ostrowski inequalities for harmonic h-preinvex functions, Facta Univ. Ser. Math. Inform., 31 (2016), no. 2, 417–445
Search in Google Scholar Back to article
[20] M. Z. Sarikaya and H. Filiz, Note on the Ostrowski type inequalities for fractional integrals, Vietnam J. Math., 42 (2014), no. 2, 187–19010.1007/s10013-014-0056-4
Search in Google Scholar Back to article
[21] M. Z. Sarikaya and H. Budak, Generalized Ostrowski type inequalities for local fractional integrals, Proc. Amer. Math. Soc., 145 (2017), no. 4, 1527–1538.10.1090/proc/13488
Search in Google Scholar Back to article
[22] 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), no. 7, 1147–1154.10.1016/j.camwa.2011.12.023
Search in Google Scholar Back to article
[23] M. Tunç, On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat, 27:4 (2013), 559–565.10.2298/FIL1304559T
Search in Google Scholar Back to article
[24] M. Tunç, Ostrowski type inequalities for m-and (α, m)-geometrically convex functions via Riemann-Louville fractional integrals, Afr. Mat., 27 (2016), no. 5-6, 841–850.10.1007/s13370-015-0378-8
Search in Google Scholar Back to article
[25] H. Yildirim and Z. Kirtay, Ostrowski inequality for generalized fractional integral and related inequalities, Malaya J. Mat., 2 (3) (2014), 322-329.10.26637/mjm203/020
Search in Google Scholar Back to article
[26] C. Yildiz, E, Özdemir and Z. S. Muhamet, New generalizations of Ostrowski-like type inequalities for fractional integrals, Kyungpook Math. J.56 (2016), no. 1, 161–172.10.5666/KMJ.2016.56.1.161
Search in Google Scholar Back to article
[27] H. Yue, Ostrowski inequality for fractional integrals and related fractional inequalities, Transylv. J. Math. Mech., 5 (2013), no. 1, 85–89.
Search in Google Scholar Back to article

Trapezoid type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation

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