Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2014, Article ID 136035, 20 pages
http://dx.doi.org/10.1155/2014/136035
Research Article

Applying GG-Convex Function to Hermite-Hadamard Inequalities Involving Hadamard Fractional Integrals

1Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China
2School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, China

Received 1 February 2014; Revised 7 June 2014; Accepted 10 June 2014; Published 14 July 2014

Academic Editor: Hari M. Srivastava

Copyright © 2014 Zhi Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. Baleanu, J. A. T. Machado, and A. C. J. Luo, Fractional Dynamics and Control, Springer, New York, NY, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  2. K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at MathSciNet
  4. V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific, 2009.
  5. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  6. M. W. Michalski, “Derivatives of noninteger order and their applications,” Dissertationes Mathematicae, vol. 328, 47 pages, 1993. View at Google Scholar
  7. I. Podlubny, Fractional Differential Equations, Academic Press, 1999. View at MathSciNet
  8. V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  9. E. Set, “New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals,” Computers & Mathematics with Applications, vol. 63, no. 7, pp. 1147–1154, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Başak, “Hermite–Hadamard's inequalities for fractional integrals and related fractional inequalities,” Mathematical and Computer Modelling, vol. 57, no. 9-10, pp. 2403–2407, 2013. View at Publisher · View at Google Scholar · View at Scopus
  11. C. Zhu, M. Fečkan, and J. Wang, “Fractional integral inequalities for differentiable convex mappings and applications to special means and a midpoint formula,” Journal of Applied Mathematics, Statistics and Informatics, vol. 8, pp. 21–28, 2012. View at Google Scholar
  12. J. Wang, X. Li, and C. Zhu, “Refinements of hermite-HADamard type inequalities involving fractional integrals,” Bulletin of the Belgian Mathematical Society—Simon Stevin, vol. 20, no. 4, pp. 655–666, 2013. View at Google Scholar · View at MathSciNet
  13. J. Wang, J. Deng, and M. Fečkan, “Hermite—hadamard-type inequalities for r-convex functions based on the use of Riemann—liouville fractional integrals,” Ukrainian Mathematical Journal, vol. 65, no. 2, pp. 193–211, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. Y. Zhang and J. Wang, “On some new Hermite-HADamard inequalities involving Riemann-Liouville fractional integrals,” Journal of Inequalities and Applications, vol. 2013, article 220, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. J. Wang, X. Li, and M. Fe.kan, “Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity,” Applicable Analysis, vol. 92, no. 11, pp. 2241–2253, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  16. J. Wang, J. Deng, and M. Fečkan, “Exploring s-e-condition and applications to some Ostrowski type inequalities via Hadamard fractional integrals,” Mathematica Slovaca. In press.
  17. J. Wang, C. Zhu, and Y. Zhou, “New generalized Hermite-HADamard type inequalities and applications to special means,” Journal of Inequalities and Applications, vol. 2013, article 325, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  18. J. Deng and J. Wang, “Fractional Hermite-Hadamard inequalities for (α,m)-logarithmically convex functions,” Journal of Inequalities and Applications, vol. 364, pp. 1–11, 2013. View at Google Scholar
  19. Y. Liao, J. Deng, and J. Wang, “Riemann-Liouville fractional Hermite-HADamard inequalities. Part I: for once differentiable geometric-arithmetically s-convex functions,” Journal of Inequalities and Applications, vol. 2013, 13 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. H. M. Srivastava, Z. H. Zhang, and Y. D. Wu, “Some further refinements and extensions of the Hermite–Hadamard and Jensen inequalities in several variables,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2709–2717, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. C. P. Niculescu, “Convexity according to means,” Mathematical Inequalities & Applications, vol. 6, no. 4, pp. 571–579, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. R. A. Satnoianu, “Improved GA-convexity inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 3, article 82, 6 pages, 2002. View at Google Scholar
  23. C. E. M. Pearce and J. Pečarić, “Inequalities for differentiable mappings with application to special means and quadrature formula,” Applied Mathematics Letters, vol. 13, no. 2, pp. 51–55, 2000. View at Publisher · View at Google Scholar · View at MathSciNet