×
Dragomir, Silvestru Sever

Trapezoid type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation. (English) Zbl 1452.26020

Acta Univ. Sapientiae, Math. 12, No. 1, 30-53 (2020).
Summary: 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.

MSC:

26D15 Inequalities for sums, series and integrals
26D10 Inequalities involving derivatives and differential and integral operators
26D07 Inequalities involving other types of functions
26A33 Fractional derivatives and integrals

Keywords:

Riemann-Liouville fractional integrals; functions of bounded variation; Lipshitzian functions; trapezoid-type inequalities
PDF BibTeX XML Cite
\textit{S. S. Dragomir}, Acta Univ. Sapientiae, Math. 12, No. 1, 30--53 (2020; Zbl 1452.26020)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] A. Aglić Aljinović, Montgomery identity and Ostrowski type inequalities for Riemann-Liouville fractional integral, J. Math., 2014, Art. ID 503195, 6 pp.
[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 · Zbl 1292.26016
[3] G. A. Anastassiou, Fractional representation formulae under initial conditions and fractional Ostrowski type inequalities, Demonstr. Math., 48 (2015), no. 3, 357-378 · Zbl 1321.26009
[4] G. A. Anastassiou, The reduction method in fractional calculus and fractional Ostrowski type inequalities, Indian J. Math., 56 (2014), no. 3, 333-357. · Zbl 1312.26012
[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.
[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. · Zbl 0966.26015
[7] S. S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comput. Math. Appl., 38 (1999), no. 11-12, 33-37. · Zbl 0974.26014
[8] S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc.60 (1999), No. 3, 495-508. · Zbl 0951.26011
[9] S. S. Dragomir, On the midpoint quadrature formula for mappings with bounded variation and applications, Kragujevac J. Math., 22 (2000), 13-19. · Zbl 1012.26016
[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] · Zbl 1016.26017
[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].
[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]. · Zbl 1358.26020
[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].
[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].
[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. · Zbl 1318.47026
[16] A. Guezane-Lakoud and F. Aissaoui, New fractional inequalities of Ostrowski type, Transylv. J. Math. Mech., 5 (2013), no. 2, 103-106 · Zbl 1442.26022
[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. · Zbl 1350.26014
[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
[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 · Zbl 1458.26063
[20] M. Z. Sarikaya and H. Filiz, Note on the Ostrowski type inequalities for fractional integrals, Vietnam J. Math., 42 (2014), no. 2, 187-190 · Zbl 1298.26026
[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. · Zbl 1357.26026
[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. · Zbl 1247.26038
[23] M. Tunç, On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat, 27:4 (2013), 559-565. · Zbl 1324.26039
[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. · Zbl 1348.26013
[25] H. Yildirim and Z. Kirtay, Ostrowski inequality for generalized fractional integral and related inequalities, Malaya J. Mat., 2 (3) (2014), 322-329. · Zbl 1371.26041
[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. · Zbl 1342.26006
[27] H. Yue, Ostrowski inequality for fractional integrals and related fractional inequalities, Transylv. J. Math. Mech., 5 (2013), no. 1, 85-89. · Zbl 1292.26067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.