Research Article
BibTex RIS Cite

Year 2020, Volume: 69 Issue: 1, 49 - 72, 30.06.2020

Sever Dragomir

https://doi.org/10.31801/cfsuasmas.542665

Abstract

References

  • Agarwal, R. P., Luo, M.-J. and Raina, R. K., On Ostrowski type inequalities, Fasc. Math. 56 (2016), 5-27.
  • Aljinović, A. Aglić, Montgomery identity and Ostrowski type inequalities for Riemann-Liouville fractional integral. J. Math. 2014, Art. ID 503195, 6 pp.
  • Apostol, T. M., Mathematical Analysis, Second Edition, Addison-Wesley Publishing Company, 1975.
  • 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
  • Anastassiou, G. A., Fractional representation formulae under initial conditions and fractional Ostrowski type inequalities. Demonstr. Math. 48 (2015), no. 3, 357--378
  • Anastassiou, G. A., The reduction method in fractional calculus and fractional Ostrowski type inequalities. Indian J. Math. 56 (2014), no. 3, 333--357.
  • 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.
  • 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.
  • Dragomir, S. S., The Ostrowski's integral inequality for Lipschitzian mappings and applications. Comput. Math. Appl. 38 (1999), no. 11-12, 33--37.
  • Dragomir, S. S., The Ostrowski integral inequality for mappings of bounded variation. Bull. Austral. Math. Soc. 60 (1999), No. 3, 495--508.
  • Dragomir, S. S., On the midpoint quadrature formula for mappings with bounded variation and applications. Kragujevac J. Math. 22 (2000), 13--19.
  • 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]
  • 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.
  • 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].
  • 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].
  • 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].
  • 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].
  • 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].
  • 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].
  • 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].
  • Guezane-Lakoud, A. and Aissaoui, F., New fractional inequalities of Ostrowski type. Transylv. J. Math. Mech. 5 (2013), no. 2, 103--106
  • 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.
  • 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.
  • 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.
  • 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
  • Raina, R. K., On generalized Wright's hypergeometric functions and fractional calculus operators, East Asian Math. J., 21(2)(2005), 191-203.
  • Sarikaya, M. Z. and Filiz, H., Note on the Ostrowski type inequalities for fractional integrals. Vietnam J. Math. 42 (2014), no. 2, 187--190
  • Sarikaya, M. Z. and Budak, H., Generalized Ostrowski type inequalities for local fractional integrals. Proc. Amer. Math. Soc. 145 (2017), no. 4, 1527--1538.
  • 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.
  • Tunç, M., On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat 27:4 (2013), 559--565.
  • 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.
  • Yildirim, H. and Kirtay, Z., Ostrowski inequality for generalized fractional integral and related inequalities, Malaya J. Mat., 2(3)(2014), 322-329.
  • 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.
  • Yue, H., Ostrowski inequality for fractional integrals and related fractional inequalities. Transylv. J. Math. Mech. 5 (2013), no. 1, 85--89.

Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation

Year 2020, Volume: 69 Issue: 1, 49 - 72, 30.06.2020

Sever Dragomir

https://doi.org/10.31801/cfsuasmas.542665

Abstract

Let g be a strictly increasing function on (a,b), having a continuous derivative g′ on (a,b). For the Lebesgue integrable function f:(a,b)→C, we define the k-g-left-sided fractional integral of f by

S_{k,g,a+}f(x)=∫_{a}^{x}k(g(x)-g(t))g′(t)f(t)dt, x∈(a,b]

and the k-g-right-sided fractional integral of f by

S_{k,g,b-}f(x)=∫_{x}^{b}k(g(t)-g(x))g′(t)f(t)dt, x∈[a,b),

where the kernel k is defined either on (0,∞) or on [0,∞) 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.

Keywords

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

References

  • Agarwal, R. P., Luo, M.-J. and Raina, R. K., On Ostrowski type inequalities, Fasc. Math. 56 (2016), 5-27.
  • Aljinović, A. Aglić, Montgomery identity and Ostrowski type inequalities for Riemann-Liouville fractional integral. J. Math. 2014, Art. ID 503195, 6 pp.
  • Apostol, T. M., Mathematical Analysis, Second Edition, Addison-Wesley Publishing Company, 1975.
  • 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
  • Anastassiou, G. A., Fractional representation formulae under initial conditions and fractional Ostrowski type inequalities. Demonstr. Math. 48 (2015), no. 3, 357--378
  • Anastassiou, G. A., The reduction method in fractional calculus and fractional Ostrowski type inequalities. Indian J. Math. 56 (2014), no. 3, 333--357.
  • 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.
  • 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.
  • Dragomir, S. S., The Ostrowski's integral inequality for Lipschitzian mappings and applications. Comput. Math. Appl. 38 (1999), no. 11-12, 33--37.
  • Dragomir, S. S., The Ostrowski integral inequality for mappings of bounded variation. Bull. Austral. Math. Soc. 60 (1999), No. 3, 495--508.
  • Dragomir, S. S., On the midpoint quadrature formula for mappings with bounded variation and applications. Kragujevac J. Math. 22 (2000), 13--19.
  • 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]
  • 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.
  • 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].
  • 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].
  • 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].
  • 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].
  • 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].
  • 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].
  • 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].
  • Guezane-Lakoud, A. and Aissaoui, F., New fractional inequalities of Ostrowski type. Transylv. J. Math. Mech. 5 (2013), no. 2, 103--106
  • 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.
  • 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.
  • 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.
  • 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
  • Raina, R. K., On generalized Wright's hypergeometric functions and fractional calculus operators, East Asian Math. J., 21(2)(2005), 191-203.
  • Sarikaya, M. Z. and Filiz, H., Note on the Ostrowski type inequalities for fractional integrals. Vietnam J. Math. 42 (2014), no. 2, 187--190
  • Sarikaya, M. Z. and Budak, H., Generalized Ostrowski type inequalities for local fractional integrals. Proc. Amer. Math. Soc. 145 (2017), no. 4, 1527--1538.
  • 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.
  • Tunç, M., On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat 27:4 (2013), 559--565.
  • 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.
  • Yildirim, H. and Kirtay, Z., Ostrowski inequality for generalized fractional integral and related inequalities, Malaya J. Mat., 2(3)(2014), 322-329.
  • 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.
  • Yue, H., Ostrowski inequality for fractional integrals and related fractional inequalities. Transylv. J. Math. Mech. 5 (2013), no. 1, 85--89.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Sever Dragomir 0000-0003-2902-6805

Publication Date June 30, 2020
Submission Date March 21, 2019
Acceptance Date July 19, 2019
Published in Issue Year 2020 Volume: 69 Issue: 1

Cite

APA Dragomir, S. (2020). Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(1), 49-72. https://doi.org/10.31801/cfsuasmas.542665
AMA Dragomir S. Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2020;69(1):49-72. doi:10.31801/cfsuasmas.542665
Chicago Dragomir, Sever. “Further Inequalities for the Generalized K-G-Fractional Integrals of Functions With Bounded Variation”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69, no. 1 (June 2020): 49-72. https://doi.org/10.31801/cfsuasmas.542665.
EndNote Dragomir S (June 1, 2020) Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69 1 49–72.
IEEE S. Dragomir, “Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 69, no. 1, pp. 49–72, 2020, doi: 10.31801/cfsuasmas.542665.
ISNAD Dragomir, Sever. “Further Inequalities for the Generalized K-G-Fractional Integrals of Functions With Bounded Variation”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69/1 (June 2020), 49-72. https://doi.org/10.31801/cfsuasmas.542665.
JAMA Dragomir S. Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69:49–72.
MLA Dragomir, Sever. “Further Inequalities for the Generalized K-G-Fractional Integrals of Functions With Bounded Variation”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 69, no. 1, 2020, pp. 49-72, doi:10.31801/cfsuasmas.542665.
Vancouver Dragomir S. Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69(1):49-72.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.