Generalized Steffensen Inequalities for Local Fractional Integrals

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Mehmet Zeki Sarikaya, Tuba Tunc, Samet Erden, Generalized Steffensen Inequalities for Local Fractional Integrals, Int. J. Anal. Appl., 14 (1) (2017), 88-98.

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Mehmet Zeki Sarikaya, Tuba Tunc, Samet Erden

Abstract

Firstly we give a important integral inequality which is generalized Steffensen's inequality. Then, we establish weighted version of generalized Steffensen's inequality for local fractional integrals. Finally, we obtain several inequalities related these inequalities using the local fractional integral.

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References

  1. S. Abramovich, M.K. Bakula, M. Matic ´, J.E. Peˇ cari ´ c, A variant of Jensen-Steffensen's inequality and quasi-arithmetic means, J. Math. Anal. Appl., 307 (2005), 370-386.
  2. J. Bergh, A generalization of Steffensen's inequality, J. Math. Anal. Appl., 41 (1973), 187-191.
  3. G-S. Chen, Generalizations of Hölder's and some related integral inequalities on fractal space, Journal of Function Spaces and Applications 2013 (2013), Article ID 198405.
  4. H. Gauchman, On a further generalization of Steffensen's inequality, J. Inequal. Appl., 5 (2000), 505-513.
  5. F. Qi, B.-N. Guo, On Steffensen pairs, J. Math. Anal. Appl. 271 (2002) 534-541.
  6. D. S. Mitrinovic, Analytic Inequalities, Springer-Verlang New-York, Heidelberg, Berlin, 1970, pp. 107-118.
  7. D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Classical and new inequalities in analysis, ser. Math. Appl. (East European Ser.). Dordrecht: Kluwer Academic Publishers Group, 1993, vol. 61, pp. 311-331.
  8. H. Mo, X Sui and D Yu, Generalized convex functions on fractal sets and two related inequalities, Abstract and Applied Analysis, 2014 (2014), Article ID 636751.
  9. H. Mo, Generalized Hermite-Hadamard inequalities involving local fractional integral, arXiv:1410.1062.
  10. J. E. Pecaric, On the Bellman generalization of Steffensen's inequality. II, J. Math. Anal. Appl. 104 (1984) 432-434.
  11. J. E. Pecaric, On the Bellman generalization of Steffensen's inequality, J. Math. Anal. Appl. 88 (1982) 505-507.
  12. M. Z. Sarikaya and H Budak, Generalized Ostrowski type inequalities for local fractional integrals, RGMIA Res. Rep. Collect. 18 (2015), Article ID 62.
  13. M. Z. Sarikaya, S.Erden and H. Budak, Some generalized Ostrowski type inequalities involving local fractional integrals and applications, RGMIA Res. Rep. Collect. 18 (2015), Article ID 63.
  14. M. Z. Sarikaya H. Budak, On generalized Hermite-Hadamard inequality for generalized convex function, RGMIA Res. Rep. Collect. 18 (2015), Article ID 64.
  15. M. Z. Sarikaya, S.Erden and H. Budak, Some integral inequalities for local fractional integrals, RGMIA Res. Rep. Collect. 18(2015), Article ID 65.
  16. M. Z. Sarikaya, H. Budak and S.Erden, On new inequalities of Simpson's type for generalized convex functions, RGMIA Res. Rep. Collect. 18(2015), Article ID 66.
  17. J. F. Steffensen, On certain inequalities and methods of approximation, J. Inst. Actuaries, 51 (1919), 274-297.
  18. U. M. Ozkan and H. Yildirim, Steffensen's integral inequality on time scales, J. Inequal. Appl. 2007 (2007), Article ID 46524.
  19. S.-H. Wu and H.M. Srivastava, Some improvements and generalizations of Steffensen's integral inequality, Appl. Math. Comput. 192 (2007) 422-428.
  20. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, 2012.
  21. J. Yang, D. Baleanu and X. J. Yang, Analysis of fractal wave equations by local fractional Fourier series method, Adv. Math. Phys. 2013 (2013), Article ID 632309.
  22. X. J. Yang, Local fractional integral equations and their applications, Adv. Comput. Sci. Appl. 1 (4) 2012, 234-239.
  23. X. J. Yang, Generalized local fractional Taylor's formula with local fractional derivative, Journal of Expert Systems, 1(1) (2012), 26-30.
  24. X. J. Yang, Local fractional Fourier analysis, Advances in Mechanical Engineering and its Applications, 1 (1) 2012, 12-16.