Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions with Bounded Variation

Sever Dragomir

doi:10.33434/cams.628097

Research Article

Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions with Bounded Variation

Year 2019, Volume: 2 Issue: 4, 309 - 330, 29.12.2019

Sever Dragomir

https://doi.org/10.33434/cams.628097

Cited By: 4

Abstract

In this paper we establish some Ostrowski and trapezoid type inequalities for the $k$-$g$-fractional integrals of functions of bounded variation. Applications for mid-point and trapezoid inequalities are provided as well. Some examples for a general exponential fractional integral are also given.

Keywords

Functions of bounded variation, Generalized Riemann-Liouville fractional integrals, Hadamard fractional integrals, Ostrowski type inequalities, Trapezoid inequalities

References

[1] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006.
[2] R. K. Raina, On generalized Wright’s hypergeometric functions and fractional calculus operators, East Asian Math. J., 21 (2) (2005), 191-203.
[3] R. P. Agarwal, M.-J. Luo, R. K. Raina, On Ostrowski type inequalities, Fasc. Math., 56 (2016), 5-27.
[4] M. Kirane, B. T. Torebek, Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via fractional integrals, Preprint arXiv:1701.00092.
[5] S. S. Dragomir, 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.
[6] A. Aglic Aljinovic, Montgomery identity and Ostrowski type inequalities for Riemann-Liouville fractional integral, J. Math., 2014, Art. ID 503195, 6 pp.
[7] T. M. Apostol, Mathematical Analysis, Second Edition, Addison-Wesley Publishing Company, 1975.
[8] A. O. Akdemir, Inequalities of Ostrowski’s type for m- and (a;m)-logarithmically convex functions via Riemann-Liouville fractional integrals, J. Comput. Anal. Appl., 16(2) (2014), 375–383.
[9] G. A. Anastassiou, Fractional representation formulae under initial conditions and fractional Ostrowski type inequalities, Demonstr. Math., 48(3) (2015), 357–378.
[10] G. A. Anastassiou, The reduction method in fractional calculus and fractional Ostrowski type inequalities, Indian J. Math., 56(3) (2014), 333-357.
[11] 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(4) (2016), 11–21.
[12] P. Cerone, 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.
[13] S. S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comput. Math. Appl., 38(11-12) (1999), 33-37.
[14] S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc., 60(3) (1999), 495-508.
[15] S. S. Dragomir, On the midpoint quadrature formula for mappings with bounded variation and applications, Kragujevac J. Math., 22 (2000), 13–19.
[16] 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.
[17] S. S. Dragomir, Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation, Arch. Math., 91(5) (2008), 450–460.
[18] 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.
[19] S. S. Dragomir, Ostrowski type inequalities for Lebesgue integral: a survey of recent results, Australian J. Math. Anal. Appl., 14(1) (2017), 1-287.
[20] 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.
[21] S. S. Dragomir, Ostrowski type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation, RGMIA Res. Rep. Coll., 20 (2017), Art. 58.
[22] A. Guezane-Lakoud, F. Aissaoui, New fractional inequalities of Ostrowski type, Transylv. J. Math. Mech., 5(2) (2013), 103–106
[23] A. Kashuri, R. Liko, Ostrowski type fractional integral inequalities for generalized (s;m;j)-preinvex functions, Aust. J. Math. Anal. Appl., 13(1) (2016), Art. 16, 11 pp.
[24] M. A. Noor, K. I. Noor, S. Iftikhar, Fractional Ostrowski inequalities for harmonic h-preinvex functions, Facta Univ. Ser. Math. Inform., 31(2) (2016), 417–445.
[25] M. Z. Sarikaya, H. Filiz, Note on the Ostrowski type inequalities for fractional integrals, Vietnam J. Math., 42(2) (2014), 187–190.
[26] M. Z. Sarikaya, H. Budak, Generalized Ostrowski type inequalities for local fractional integrals, Proc. Amer. Math. Soc., 145(4) (2017), 1527–1538.
[27] 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(7) (2012), 1147-1154.
[28] M. Tunç, On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat, 27(4) (2013), 559–565.
[29] M. Tunç, Ostrowski type inequalities for m- and (a;m)-geometrically convex functions via Riemann-Louville fractional integrals, Afr. Mat., 27(5-6) (2016), 841–850.
[30] H. Yildirim, Z. Kirtay, Ostrowski inequality for generalized fractional integral and related inequalities, Malaya J. Mat., 2(3) (2014), 322-329.
[31] C. Yildiz, E, Özdemir, Z. S. Muhamet, New generalizations of Ostrowski-like type inequalities for fractional integrals, Kyungpook Math. J., 56(1) (2016), 161–172.
[32] H. Yue, Ostrowski inequality for fractional integrals and related fractional inequalities, Transylv. J. Math. Mech., 5(1) (2013), 85–89.

Year 2019, Volume: 2 Issue: 4, 309 - 330, 29.12.2019

Sever Dragomir

https://doi.org/10.33434/cams.628097

Cited By: 4

Abstract

References

[1] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006.
[2] R. K. Raina, On generalized Wright’s hypergeometric functions and fractional calculus operators, East Asian Math. J., 21 (2) (2005), 191-203.
[3] R. P. Agarwal, M.-J. Luo, R. K. Raina, On Ostrowski type inequalities, Fasc. Math., 56 (2016), 5-27.
[4] M. Kirane, B. T. Torebek, Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via fractional integrals, Preprint arXiv:1701.00092.
[5] S. S. Dragomir, 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.
[6] A. Aglic Aljinovic, Montgomery identity and Ostrowski type inequalities for Riemann-Liouville fractional integral, J. Math., 2014, Art. ID 503195, 6 pp.
[7] T. M. Apostol, Mathematical Analysis, Second Edition, Addison-Wesley Publishing Company, 1975.
[8] A. O. Akdemir, Inequalities of Ostrowski’s type for m- and (a;m)-logarithmically convex functions via Riemann-Liouville fractional integrals, J. Comput. Anal. Appl., 16(2) (2014), 375–383.
[9] G. A. Anastassiou, Fractional representation formulae under initial conditions and fractional Ostrowski type inequalities, Demonstr. Math., 48(3) (2015), 357–378.
[10] G. A. Anastassiou, The reduction method in fractional calculus and fractional Ostrowski type inequalities, Indian J. Math., 56(3) (2014), 333-357.
[11] 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(4) (2016), 11–21.
[12] P. Cerone, 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.
[13] S. S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comput. Math. Appl., 38(11-12) (1999), 33-37.
[14] S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc., 60(3) (1999), 495-508.
[15] S. S. Dragomir, On the midpoint quadrature formula for mappings with bounded variation and applications, Kragujevac J. Math., 22 (2000), 13–19.
[16] 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.
[17] S. S. Dragomir, Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation, Arch. Math., 91(5) (2008), 450–460.
[18] 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.
[19] S. S. Dragomir, Ostrowski type inequalities for Lebesgue integral: a survey of recent results, Australian J. Math. Anal. Appl., 14(1) (2017), 1-287.
[20] 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.
[21] S. S. Dragomir, Ostrowski type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation, RGMIA Res. Rep. Coll., 20 (2017), Art. 58.
[22] A. Guezane-Lakoud, F. Aissaoui, New fractional inequalities of Ostrowski type, Transylv. J. Math. Mech., 5(2) (2013), 103–106
[23] A. Kashuri, R. Liko, Ostrowski type fractional integral inequalities for generalized (s;m;j)-preinvex functions, Aust. J. Math. Anal. Appl., 13(1) (2016), Art. 16, 11 pp.
[24] M. A. Noor, K. I. Noor, S. Iftikhar, Fractional Ostrowski inequalities for harmonic h-preinvex functions, Facta Univ. Ser. Math. Inform., 31(2) (2016), 417–445.
[25] M. Z. Sarikaya, H. Filiz, Note on the Ostrowski type inequalities for fractional integrals, Vietnam J. Math., 42(2) (2014), 187–190.
[26] M. Z. Sarikaya, H. Budak, Generalized Ostrowski type inequalities for local fractional integrals, Proc. Amer. Math. Soc., 145(4) (2017), 1527–1538.
[27] 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(7) (2012), 1147-1154.
[28] M. Tunç, On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat, 27(4) (2013), 559–565.
[29] M. Tunç, Ostrowski type inequalities for m- and (a;m)-geometrically convex functions via Riemann-Louville fractional integrals, Afr. Mat., 27(5-6) (2016), 841–850.
[30] H. Yildirim, Z. Kirtay, Ostrowski inequality for generalized fractional integral and related inequalities, Malaya J. Mat., 2(3) (2014), 322-329.
[31] C. Yildiz, E, Özdemir, Z. S. Muhamet, New generalizations of Ostrowski-like type inequalities for fractional integrals, Kyungpook Math. J., 56(1) (2016), 161–172.
[32] H. Yue, Ostrowski inequality for fractional integrals and related fractional inequalities, Transylv. J. Math. Mech., 5(1) (2013), 85–89.

There are 32 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Articles
Authors	Sever Dragomir 0000-0003-2902-6805
Publication Date	December 29, 2019
Submission Date	October 2, 2019
Acceptance Date	November 21, 2019
Published in Issue	Year 2019 Volume: 2 Issue: 4

Cite

APA	Dragomir, S. (2019). Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions with Bounded Variation. Communications in Advanced Mathematical Sciences, 2(4), 309-330. https://doi.org/10.33434/cams.628097
AMA	Dragomir S. Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions with Bounded Variation. Communications in Advanced Mathematical Sciences. December 2019;2(4):309-330. doi:10.33434/cams.628097
Chicago	Dragomir, Sever. “Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions With Bounded Variation”. Communications in Advanced Mathematical Sciences 2, no. 4 (December 2019): 309-30. https://doi.org/10.33434/cams.628097.
EndNote	Dragomir S (December 1, 2019) Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions with Bounded Variation. Communications in Advanced Mathematical Sciences 2 4 309–330.
IEEE	S. Dragomir, “Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions with Bounded Variation”, Communications in Advanced Mathematical Sciences, vol. 2, no. 4, pp. 309–330, 2019, doi: 10.33434/cams.628097.
ISNAD	Dragomir, Sever. “Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions With Bounded Variation”. Communications in Advanced Mathematical Sciences 2/4 (December 2019), 309-330. https://doi.org/10.33434/cams.628097.
JAMA	Dragomir S. Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions with Bounded Variation. Communications in Advanced Mathematical Sciences. 2019;2:309–330.
MLA	Dragomir, Sever. “Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions With Bounded Variation”. Communications in Advanced Mathematical Sciences, vol. 2, no. 4, 2019, pp. 309-30, doi:10.33434/cams.628097.
Vancouver	Dragomir S. Ostrowski and Trapezoid Type Inequalities for the Generalized $k$-$g$-Fractional Integrals of Functions with Bounded Variation. Communications in Advanced Mathematical Sciences. 2019;2(4):309-30.