Table of Contents
ISRN Mathematical Analysis
Volume 2013, Article ID 903196, 9 pages
http://dx.doi.org/10.1155/2013/903196
Research Article

Hardy-Type Inequalities on Time Scale via Convexity in Several Variables

1Department of Mathematics, University Al. I. Cuza, 700506 Iaşi, Romania
2Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan
3University of Zagreb, Faculty of Textile Technology, 10000 Zagreb, Croatia

Received 3 June 2013; Accepted 1 July 2013

Academic Editors: M. Lindstrom and S. Liu

Copyright © 2013 Tzanko Donchev 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. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 1988, Reprint of the 1952 edition. View at MathSciNet
  2. E. K. Godunova, “Inequalities based on convex functions,” Izvestija Vysših Učebnyh Zavedeniĭ Matematika, vol. 1965, no. 4, pp. 45–53, 1965. View at Google Scholar · View at MathSciNet
  3. S. Kaijser, L. Nikolova, L.-E. Persson, and A. Wedestig, “Hardy-type inequalities via convexity,” Mathematical Inequalities & Applications, vol. 8, no. 3, pp. 403–417, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. S. Kaijser, L.-E. Persson, and A. Öberg, “On Carleman and Knopp's inequalities,” Journal of Approximation Theory, vol. 117, no. 1, pp. 140–151, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. K. Krulić, J. Pečarić, and L.-E. Persson, “Some new Hardy type inequalities with general kernels,” Mathematical Inequalities & Applications, vol. 12, no. 3, pp. 473–485, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, vol. 53 of Mathematics and its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1991. View at MathSciNet
  7. U. M. Ozkan and H. Yildirim, “Hardy-Knopp-type inequalities on time scales,” Dynamic Systems and Applications, vol. 17, no. 3-4, pp. 477–486, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. U. M. Özkan and H. Yıldırım, “Time scale Hardy-Knopp type integral inequalities,” Communications in Mathematical Analysis, vol. 6, no. 1, pp. 36–41, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. P. Řehák, “Hardy inequality on time scales and its application to half-linear dynamic equations,” Journal of Inequalities and Applications, no. 5, pp. 495–507, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  10. M. Bohner, A. Nosheen, J. E. Pečarić, and A. Younus, “Some dynamic Hardy-type inequalities with general kernel,” Journal of Mathematical Inequalities. In press.
  11. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  12. M. Bohner and G. Sh. Guseinov, “Multiple integration on time scales,” Dynamic Systems and Applications, vol. 14, no. 3-4, pp. 579–606, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. Bohner and G. Sh. Guseinov, “Multiple Lebesgue integration on time scales,” Advances in Difference Equations, vol. 2006, Article ID 26391, 12 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. G. Sh. Guseinov, “Integration on time scales,” Journal of Mathematical Analysis and Applications, vol. 285, no. 1, pp. 107–127, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, NY, USA, 3rd edition, 1988. View at MathSciNet
  16. R. Bibi, M. Bohner, J. E. Pečarić, and S. Vorošanec, “Minkoski and Beckenback-Oresher inequalities and functionals on time scales,” Journal of Mathematical Inequalities. In press.
  17. M. Anwar, R. Bibi, M. Bohner, and J. E. Pečarić, “Jensen's functional on time scales for multi-variables,” submitted.
  18. P. M. Vasić and J. E. Pečarić, “Notes on some inequalities for convex functions,” Matematički Vesnik, vol. 6, no. 19, pp. 185–193, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. E. Beck, “Über Ungleichungen von der Form f(Mφ(x;α), Mψ(y;α))Mχ(f(x,y);α),” Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, vol. 320–328, pp. 1–14, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, 1993. View at MathSciNet
  21. L. Horváth, Kh. Ali Khan, and J. Pečarić, “Refinements of Hölder and Minkowski inequalities with weights,” Proceedings of A. Razmadze Mathematical Institute, vol. 158, pp. 33–56, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. L. Horváth, K. A. Khan, and J. E. Pečarić, “Refinements of Hölder and Minkowski inequalities with weights,” Proceedings of A. Razmadze Mathematical Institute, vol. 158, pp. 33–56, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. Krnić, N. Lovričević, and J. E. Pečarić, “On the properties of McShane’s functional and their applications,” Periodica Mathematica Hungarica, vol. 66, no. 2, pp. 159–180, 2013. View at Publisher · View at Google Scholar
  24. R. P. Boas, Jr. and C. O. Imoru, “Elementary convolution inequalities,” SIAM Journal on Mathematical Analysis, vol. 6, pp. 457–471, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. Anwar, R. Bibi, M. Bohner, and J. Pečarić, “Integral inequalities on time scales via the theory of isotonic linear functionals,” Abstract and Applied Analysis, vol. 2011, Article ID 483595, 16 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet