Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

NEW FRACTIONAL INTEGRAL INEQUALITIES OF TYPE OSTROWSKI THROUGH GENERALIZED CONVEX FUNCTION

  • HUSSAIN, SABIR (Department of Mathematics, College of Science, Qassim University) ;
  • QAISAR, SHAHID (Department of Mathematics, Comsats Institute of Information Technology Sahiwal)
  • Received : 2017.08.24
  • Accepted : 2017.12.10
  • Published : 2018.01.30
Download PDF
⟨ Previous Next ⟩

Abstract

We establish some new ostrowski type inequalities for MT-convex function including first order derivative via Niemann-Trouvaille fractional integral. It is interesting to mention that our results provide new estimates on these types of integral inequalities for MT-convex functions.

Keywords

References

  1. A.M. Ostrowski, Uber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Commentarii Mathematici Helvetici 10 (1938), 226-227.
  2. D.S. Mitrinovic, J.E. Pecaric and A.M. Fink, Inequalities involving functions and their integrals and derivatives 53, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.
  3. P. Cerone and S.S. Dragomir, Trapezoidal-type rules from an inequalities point of view, in Handbook of Analytic-Computational Methods in Applied Mathematics, pp. 65-134, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2000.
  4. J. Duoandikoetxea, A unified approach to several inequalities involving functions and derivatives, Czechoslovak Mathematical Journal 51 (2001), 363-376. https://doi.org/10.1023/A:1013703215722
  5. S.S. Dragomir and N.S. Barnett, An ostrowski type inequality for mappings whose second derivatives are bounded and applications, RGMIA Research Report Collection 1 (1999), 67-76.
  6. S.S. Dragomir, An ostrowski type inequality for convex functions, Univerzitet u Beogradu Publikacije Elektrotehnickog Fakulteta Serija Matematika 16 (2005), 12-25.
  7. Z. Liu, Some companions of an ostrowski type inequality and applications, Journal of Inequalities in Pure and Applied Mathematics 10 (2009), Article 52, 12 pages.
  8. M.Z. Sarikaya, On the ostrowski type integral inequality, Acta Mathematica Universitatis Comenianae 79 (2010), 129-134.
  9. M.Z. Sarikaya, On the ostrowski type integral inequality for double integrals, Demonstratio Mathematica 45 (2012), 533-540.
  10. M.Z. Sarikaya and H. Ogunmez, On the weighted ostrowski-type integral inequality for double integrals, Arabian Journal for Science and Engineering 36 (2011), 1153-1160. https://doi.org/10.1007/s13369-011-0102-4
  11. R. Gorenflo and F. Mainardi, Fractional calculus: Integral and Differentiable Equations of Fractional Order, Springer, Wien, Austria, 1997.
  12. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives Theory and Application, Gordan and Breach Science, New York, NY, USA, 1993.
  13. G. Anastassiou, M.R. Hooshmandasl, A. Ghasemi and F. Moftakharzadeh, Montgomery identities for fractional integrals and related fractional inequalities, Journal of Inequalities in Pure and Applied Mathematics 10 (2009), Article 97, 6 pages.
  14. S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, Journal of Inequalities in Pure and Applied Mathematics 10 (2009), Article 86, 5 pages.
  15. Z. Dahmani, L. Tabharit and S. Taf, Some fractional integral inequalities, Nonlinear Science Letters 2 (2010), 155-160.
  16. Z. Dahmani, L. Tabharit and S. Taf, New inequalities via Niemann-Trouvaille fractional integration, Journal of Advanced Research in Scientific Computing 2 (2010), 40-45.
  17. K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 2010.
  18. Z. Dahmani, New inequalities in fractional integrals, International Journal of Nonlinear Science 9 (2010), 493-497.
  19. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  20. M.Z. Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model. 57(2013), 2403-2407. https://doi.org/10.1016/j.mcm.2011.12.048
  21. M. Tunc and H. Yildirim, On MT-convexity, arXiv:1205.5453v1 [math.CA].
  22. S. Qaisar, C. He and S. Hussain, On New Inequalities of Hermite-Hadamard type for generalized Convex Functions, Italian journal of Pure and Applied Mathematics 33 (2014), 139-148.
  23. S. Qaisar and S. Hussain, Some results on Hermite-Hadamard type inequality through convexity. Turkish J. Anal. Num. Theo. 2(2) (2014), 53-59. https://doi.org/10.12691/tjant-2-2-5
  24. S. Qaisar, C. He and S. Hussain, New integral inequalities through invexity with applications, International Journal of Analysis and Applications 5 (2014), 115-122.
  25. S. Qaisar, C. He and S. Hussain, A generalization of Simpson's type inequality for differentiable functions using (${\alpha}$, m)- convex function and applications, Journal of Inequalities and Applications 158 (2013), 13 pages.
  26. S. Hussain and S. Qaisar, Generalization of Simpson's type inequality through preinvexity and prequasiinvexity, Punjab University Journal of Mathematics 46 (2014), 1-9.