Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 198405, 9 pages
http://dx.doi.org/10.1155/2013/198405
Research Article

Generalizations of Hölder’s and Some Related Integral Inequalities on Fractal Space

Department of Construction and Information Engineering, Guangxi Modern Vocational Technology College, Hechi, Guangxi 547000, China

Received 5 May 2013; Accepted 8 July 2013

Academic Editor: Miguel Martin

Copyright © 2013 Guang-Sheng Chen. 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. E. Hewitt and K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Second Printing Corrected, Springer, New York, NY, USA, 1969. View at Zentralblatt MATH · View at MathSciNet
  2. J. Kuang, Applied Inequalities, Shandong Science Press, Jinan, China, 2003.
  3. G. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1953.
  4. X. Yang, “A generalization of Hölder inequality,” Journal of Mathematical Analysis and Applications, vol. 247, no. 1, pp. 328–330, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. Yang, “Refinement of Hölder inequality and application to Ostrowski inequality,” Applied Mathematics and Computation, vol. 138, no. 2-3, pp. 455–461, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. X. Yang, “A note on Hölder inequality,” Applied Mathematics and Computation, vol. 134, no. 2-3, pp. 319–323, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. X. Yang, “Hölder's inequality,” Applied Mathematics Letters, vol. 16, no. 6, pp. 897–903, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Wu and L. Debnath, “Generalizations of Aczél's inequality and Popoviciu's inequality,” Indian Journal of Pure and Applied Mathematics, vol. 36, no. 2, pp. 49–63, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. H. Wu, “Generalization of a sharp Hölder's inequality and its application,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 741–750, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. Wu, “A new sharpened and generalized version of Hölder's inequality and its applications,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 708–714, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. E. G. Kwon and E. K. Bae, “On a continuous form of Hölder inequality,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 585–593, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. X. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic publisher Limited, Hong Kong, 2011.
  13. X. J. Yang, “Local fractional integral transforms,” Progress in Nonlinear Science, vol. 4, pp. 1–225, 2011. View at Google Scholar
  14. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2013.
  15. W.-H. Su, D. Baleanu, X.-J. Yang, and H. Jafari, “Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method,” Fixed Point Theory and Applications, vol. 2013, no. 1, pp. 89–102, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  16. M.-S. Hu, D. Baleanu, and X.-J. Yang, “One-phase problems for discontinuous heat transfer in fractal media,” Mathematical Problems in Engineering, vol. 2013, Article ID 358473, 3 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  17. K. M. Kolwankar and A. D. Gangal, “Hölder exponents of irregular signals and local fractional derivatives,” Pramana, vol. 48, no. 1, pp. 49–68, 1997. View at Google Scholar · View at Scopus
  18. K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2, pp. 214–217, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. G. Jumarie, “The Minkowski's space-time is consistent with differential geometry of fractional order,” Physics Letters A, vol. 363, no. 1-2, pp. 5–11, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. A. Parvate and A. D. Gangal, “Calculus on fractal subsets of real line. I. Formulation,” Fractals, vol. 17, no. 1, pp. 53–81, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. Parvate and A. D. Gangal, “Fractal differential equations and fractal-time dynamical systems,” Pramana, vol. 64, no. 3, pp. 389–409, 2005. View at Google Scholar · View at Scopus
  23. W. Chen, “Time-space fabric underlying anomalous diffusion,” Chaos, Solitons and Fractals, vol. 28, no. 4, pp. 923–929, 2006. View at Publisher · View at Google Scholar · View at Scopus
  24. W. Chen, X. D. Zhang, and D. Korošak, “Investigation on fractional and fractal derivative relaxation-oscillation models,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 1, pp. 3–9, 2010. View at Google Scholar · View at Scopus
  25. W. Chen, H. Sun, X. Zhang, and D. Korošak, “Anomalous diffusion modeling by fractal and fractional derivatives,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1754–1758, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. F. B. Adda and J. Cresson, “About non-differentiable functions,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 721–737, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. H. He, “A new fractal derivation,” Thermal Science, vol. 15, no. 1, pp. 145–147, 2011. View at Google Scholar
  28. J.-H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. Fan and J. He, “Fractal derivative model for air permeability in hierarchic porous media,” Abstract and Applied Analysis, vol. 2012, Article ID 354701, 7 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Y. J. Yang, D. Baleanu, and X. J. Yang, “A local fractional variational iteration method for Laplace equation within local fractional operators,” Abstract and Applied Analysis, vol. 2013, Article ID 202650, 6 pages, 2013. View at Publisher · View at Google Scholar
  31. F. Gao, X. Yang, and Z. Kang, “Local fractional newton's method derived from modified local fractional calculus,” in Proceedings of the 2nd International Joint Conference on Computational Sciences and Optimization (CSO '09), pp. 228–232, IEEE Computer Society, April 2009. View at Publisher · View at Google Scholar · View at Scopus
  32. X. Yang and F. Gao, “The fundamentals of local fractional derivative of the one-variable nondifferentiable functions,” Science & Technology, vol. 31, no. 5, pp. 920–921, 2009. View at Google Scholar
  33. X. Yang and F. Gao, “Fundamentals of Local fractional iteration of the continuously non-differentiable functions derived from local fractional calculus,” in Proceedings of the 2011 International Conference on Computer Science and Information Engineering (CSIE '11), pp. 398–404, Springer, 2011.
  34. X. Yang, L. Li, and R. Yang, “Problems of local fractional definite integral of the one-variable nondifferentiable function,” Science & Technology, vol. 31, no. 4, pp. 722–724, 2009. View at Google Scholar
  35. X. Yang, L. Li, and R. Yang, “Problems of local fractional definite integral of the one-variable nondifferentiable function,” Science & Technology, vol. 31, no. 4, pp. 722–724, 2009. View at Google Scholar
  36. W.-H. Su, X.-J. Yang, H. Jafari, and D. Baleanu, “Fractional complex transform method for wave equations on Cantor sets within local fractional differential operator,” Advances in Difference Equations, vol. 2013, no. 1, pp. 97–107, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  37. G. S. Chen, “The local fractional Stieltjes Transform in fractal space,” Advances in Intelligent Transportation Systems, vol. 1, no. 1, pp. 29–31, 2012. View at Google Scholar
  38. G. S. Chen, “Local fractional Improper integral in fractal space,” Advances in Information Technology and Management, vol. 1, no. 1, pp. 4–8, 2012. View at Google Scholar
  39. G. S. Chen, “Mean value theorems for local fractional integrals on fractal space,” Advances in Mechanical Engineering and Its Applications, vol. 1, no. 1, pp. 5–8, 2012. View at Google Scholar