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Arabian Journal of Mathematics

, Volume 8, Issue 2, pp 95–107 | Cite as

Local fractional integrals involving generalized strongly m-convex mappings

  • George AnastassiouEmail author
  • Artion Kashuri
  • Rozana Liko
Open Access
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Abstract

In this paper, we first obtain a generalized integral identity for twice local fractional differentiable mappings on fractal sets \({\mathbb {R}}^{\alpha }\, (0<\alpha \le 1)\) of real line numbers. Then, using twice local fractional differentiable mappings that are in absolute value at certain powers generalized strongly m-convex, we obtain some new estimates on generalization of trapezium-like inequalities. We also discuss some new special cases which can be deduced from our main results.

Mathematics Subject Classification

Primary 26A51 Secondary 26A33 26D07 26D10 26D15 

Notes

References

  1. 1.
    Agarwal, P.: Some inequalities involving Hadamard type \(k\)-fractional integral operators. Math. Methods Appl. Sci. 40, 3882–3891 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Agarwal, P.; Jleli, M.; Tomar, M.: Certain Hermite-Hadamard type inequalities via generalized \(k\)-fractional integrals. J. Inequal. Appl. 2017, 10 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anastassiou, G.: Fractional Differentiation Inequalities. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Anastassiou, G.: Advances on Fractional Inequalities. Springer, New York (2011)CrossRefGoogle Scholar
  5. 5.
    Anastassiou, G.: Intelligent Mathematics: Computational Analysis. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Budak, H.; Sarikaya, M.Z.; Yildirim, H.: New inequalities for local fractional integrals. RGMIA Res. Rep. Collect. 18, 13 (2015)zbMATHGoogle Scholar
  7. 7.
    Choi, J.-S.; Set, E.; Tomar, M.: Certain generalized Ostrowski type inequalities for local fractional integrals. Commun. Korean Math. Soc. 32, 601–617 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Erden, S.; Sarikaya, M.Z.: Generalized Pompeiu type inequalities for local fractional integrals and its applications. Appl. Math. Comput. 274, 282–291 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. Appl. Math. Lett. 22, 378–385 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lara, T.; Merentes, N.; Quintero, R.; Rosales, E.: On strongly \(m\)-convex functions. Math. Aeterna 5(3), 521–535 (2015)Google Scholar
  11. 11.
    Lara, T.; Merentes, N.; Quintero, R.: On inequalities of Fejér and Hermite-Hadamard types for strongly \(m\)-convex functions. Math. Aeterna 5(5), 777–793 (2015)Google Scholar
  12. 12.
    Mo, H.-X.: Generalized Hermite-Hadamard inequalities involving local fractional integral. Arxiv 2014, 8 (2014)Google Scholar
  13. 13.
    Mo, H.-X.; Sui, X.: Generalized \(s\)-convex functions on fractal sets. Abstr. Appl. Anal. 2014, 8 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mo, H.-X.; Sui, X.: Hermite-Hadamard type inequalities for generalized \(s\)-convex functions on real linear fractal set \({\mathbb{R}}^{\alpha }\, (0<\alpha \le 1)\). Math. Sci. (Springer) 11, 241–246 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mo, H.-X.; Sui, X.; Yu, D.-Y.: Generalized convex functions on fractal sets and two related inequalities. Abstr. Appl. Anal. 2014, 7 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Sarikaya, M.Z.; Budak, H.: Generalized Ostrowski type inequalities for local fractional integrals. Proc. Am. Math. Soc. 145, 1527–1538 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Sarikaya, M.Z.; Erden, S.; Budak, H.: Some generalized Ostrowski type inequalities involving local fractional integrals and applications. RGMIA Res. Rep. Collect. 18, 12 (2015)Google Scholar
  18. 18.
    Sarikaya, M.Z.; Tunç, M.; Budak, H.: On generalized some integral inequalities for local fractional integrals. Appl. Math. Comput. 276, 316–323 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Set, E.; Tomar, M.: New inequalities of Hermite-Hadamard type for generalized convex functions with applications. Facta Univ. Ser. Math. Inform. 31, 383–397 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Srivastava, H.M.; Choi, J.-S.: Zeta and \(q\)-Zeta functions and associated series and integrals. Elsevier Inc, Amsterdam (2012)zbMATHGoogle Scholar
  21. 21.
    Tomar, M.; Agarwal, P.; Jleli, M.; Samet, B.: Certain Ostrowski type inequalities for generalized \(s\)-convex functions. J. Nonlinear Sci. Appl. 10, 5947–5957 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yang, X.-J.: Generalized local fractional Taylor’s formula with local fractional derivative. Arxiv 2011, 5 (2011)Google Scholar
  23. 23.
    Yang, X.-J.: Local fractional functional analysis and its applications. Asian Academic publisher Limited, Hong Kong (2011)Google Scholar
  24. 24.
    Yang, X.-J.: Advanced Local Fractional Calculus and its Applications. World Science Publisher, New York (2012)Google Scholar
  25. 25.
    Yang, X.-J.: Local fractional Fourier analysis. Adv. Mech. Eng. Appl. 1, 12–16 (2012)Google Scholar
  26. 26.
    Yang, X.-J.: Local fractional integral equations and their applications. Adv. Comput. Sci. Appl. 1, 234–239 (2012)Google Scholar
  27. 27.
    Yang, Y.-J.; Baleanu, D.; Yang, X.-J.: Analysis of fractal wave equations by local fractional Fourier series method. Adv. Math. Phys. 2013, 6 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • George Anastassiou
    • 1
    Email author
  • Artion Kashuri
    • 2
  • Rozana Liko
    • 2
  1. 1.George Anastassiou Department of Mathematical SciencesUniversity of MemphisMemphisUSA
  2. 2.Department of Mathematics, Faculty of Technical ScienceUniversity Ismail QemaliVloraAlbania

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