## Further inequalities for the generalized $$k$$-$$g$$-fractional integrals of functions with bounded variation.(English)Zbl 1492.26026

Summary: Let $$g$$ be a strictly increasing function on $$(a,b)$$, having a continuous derivative $$g^\prime$$ on $$(a,b)$$. For the Lebesgue integrable function $$f:(a,b)\to \mathbb{C}$$, we define the $$k$$-$$g$$-left-sided fractional integral of $$f$$ by $S_{k,g,a+}f(x)=\int_a^x k(g(x)-g(t))g^\prime(t)f(t)dt, \, x \in (a,b]$ and the $$k$$-$$g$$-right-sided fractional integral of $$f$$ by $S_{k,g,b-}f(x)= \int_x^b k(g(t)-g(x))g^\prime(t)f(t)dt, \, x \in [a,b),$ where the kernel $$k$$ is defined either on $$(0, \infty )$$ or on $$[0, \infty )$$ with complex values and integrable on any finite subinterval.
In this paper we establish some new inequalities for the $$k$$-$$g$$-fractional integrals of functions of bounded variation. Examples for the generalized left- and right-sided Riemann-Liouville fractional integrals of a function f with respect to another function g and a general exponential fractional integral are also provided.

### MSC:

 26D15 Inequalities for sums, series and integrals 26A33 Fractional derivatives and integrals 26A45 Functions of bounded variation, generalizations 26D10 Inequalities involving derivatives and differential and integral operators
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### References:

 [1] Agarwal, R. P., Luo, M.-J. and Raina, R. K., On Ostrowski type inequalities, Fasc. Math. 56 (2016), 5-27. · Zbl 1352.26003 [2] Aljinović, A. Aglić, Montgomery identity and Ostrowski type inequalities for Riemann-Liouville fractional integral. J. Math. 2014, Art. ID 503195, 6 pp. · Zbl 1489.26023 [3] Apostol, T. M., Mathematical Analysis, Second Edition, Addison-Wesley Publishing Company, 1975. [4] Akdemir, A. O., 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 [5] Anastassiou, G. A., Fractional representation formulae under initial conditions and fractional Ostrowski type inequalities. Demonstr. Math. 48 (2015), no. 3, 357-378 · Zbl 1321.26009 [6] Anastassiou, G. A., The reduction method in fractional calculus and fractional Ostrowski type inequalities. Indian J. Math. 56 (2014), no. 3, 333-357. · Zbl 1312.26012 [7] Budak, H., Sarikaya, M. Z. and Set, E., 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. · Zbl 1462.26023 [8] Cerone, P. and Dragomir, S. S., 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 [9] Dragomir, S. S., The Ostrowski’s integral inequality for Lipschitzian mappings and applications. Comput. Math. Appl. 38 (1999), no. 11-12, 33-37. · Zbl 0974.26014 [10] Dragomir, S. S., The Ostrowski integral inequality for mappings of bounded variation. Bull. Austral. Math. Soc. 60 (1999), No. 3, 495-508. · Zbl 0951.26011 [11] Dragomir, S. S., On the midpoint quadrature formula for mappings with bounded variation and applications. Kragujevac J. Math. 22 (2000), 13-19. · Zbl 1012.26016 [12] Dragomir, S. S., 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 [13] Dragomir, S. S., Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation. Arch. Math. (Basel) 91 (2008), no. 5, 450-460. · Zbl 1162.26005 [14] Dragomir, S. S., 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]. · Zbl 1296.26070 [15] Dragomir, S. S., 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 [16] Dragomir, S. S., 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]. [17] Dragomir, S. S., Ostrowski type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation, RGMIA Res. Rep. Coll. 20 (2017), Art. 58. [Online http://rgmia.org/papers/v20/v20a58.pdf]. [18] Dragomir, S. S., Further Ostrowski and trapezoid type inequalities for the generalized Riemann-Liouville fractional integrals of functions with bounded variation, RGMIA Res. Rep. Coll. 20 (2017), Art. 84. [Online http://rgmia.org/papers/v20/v20a84.pdf]. [19] Dragomir, S. S., Ostrowski and trapezoid type inequalities for the generalized k-g-fractional integrals of functions with bounded variation, RGMIA Res. Rep. Coll. 20 (2017), Art. 111. [Online http://rgmia.org/papers/v20/v20a111.pdf]. [20] Dragomir, S. S., Some inequalities for the generalized k-g-fractional integrals of functions under complex boundedness conditions, RGMIA Res. Rep. Coll. 20 (2017), Art. 119. [Online http://rgmia.org/papers/v20/v20a119.pdf]. [21] Guezane-Lakoud, A. and Aissaoui, F., New fractional inequalities of Ostrowski type. Transylv. J. Math. Mech. 5 (2013), no. 2, 103-106 · Zbl 1442.26022 [22] Kashuri, A. and Liko, R., 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 [23] Kilbas, A., Srivastava, H. M. and Trujillo, J. J., 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. · Zbl 1092.45003 [24] Kirane, M. and Torebek, B. T., Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte type Inequalities for convex functions via fractional integrals, Preprint arXiv:1701.00092. [25] Noor, M. A., Noor, K. I. and Iftikhar, S., Fractional Ostrowski inequalities for harmonic h-preinvex functions. Facta Univ. Ser. Math. Inform. 31 (2016), no. 2, 417-445 · Zbl 1458.26063 [26] Raina, R. K., On generalized Wright’s hypergeometric functions and fractional calculus operators, East Asian Math. J., 21(2)(2005), 191-203. · Zbl 1101.33016 [27] Sarikaya, M. Z. and Filiz, H., Note on the Ostrowski type inequalities for fractional integrals. Vietnam J. Math. 42 (2014), no. 2, 187-190 · Zbl 1298.26026 [28] Sarikaya, M. Z. and Budak, H., Generalized Ostrowski type inequalities for local fractional integrals. Proc. Amer. Math. Soc. 145 (2017), no. 4, 1527-1538. · Zbl 1357.26026 [29] Set, E., 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 [30] Tunç, M., On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat 27:4 (2013), 559-565. · Zbl 1324.26039 [31] Tunç, M., 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 [32] Yildirim, H. and Kirtay, Z., Ostrowski inequality for generalized fractional integral and related inequalities, Malaya J. Mat., 2(3)(2014), 322-329. · Zbl 1371.26041 [33] Yildiz, C., Özdemir, E and Muhamet, Z. S., New generalizations of Ostrowski-like type inequalities for fractional integrals. Kyungpook Math. J. 56 (2016), no. 1, 161-172. · Zbl 1342.26006 [34] Yue, H., Ostrowski inequality for fractional integrals and related fractional inequalities. Transylv. J. Math. Mech. 5 (2013), no. 1, 85-89. · Zbl 1292.26067
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