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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 258108, 12 pages
http://dx.doi.org/10.1155/2014/258108
Research Article

Schur -Power Convexity of a Class of Multiplicatively Convex Functions and Applications

1School of Mathematics and Statistics, Hefei Normal University, Hefei 230601, China
2Department of Mathematics and Physics, Anhui Xinhua University, Hefei 230088, China

Received 14 January 2014; Revised 2 April 2014; Accepted 23 April 2014; Published 20 May 2014

Academic Editor: Marco Sabatini

Copyright © 2014 Wen Wang and Shiguo Yang. 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. P. S. Bullen, D. S. Mitrinovic, and M. Vasic, Handbook of Means and Theirs Inequality, Kluwer Academic, Dordrecht, The Netherlands, 2003.
  2. T. Hara, M. Uchiyama, and S.-E. Takahasi, “A refinement of various mean inequalities,” Journal of Inequalities and Applications, vol. 2, no. 4, pp. 387–395, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. K. Z. Guan, “A class of symmetric functions for multiplicatively convex function,” Mathematical Inequalities & Applications, vol. 10, no. 4, pp. 745–753, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. I. Rovenţa, “Schur convexity of a class of symmetric functions,” Annals of the University of Craiova. Mathematics and Computer Science Series, vol. 37, no. 1, pp. 12–18, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. X. Meng, Y. M. Chu, and X. M. Tang, “The Schur-harmonic-convexity of dual form of the Hamy symmetric function,” Matematichki Vesnik, vol. 62, no. 1, pp. 37–46, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. N. Shi and J. Zhang, “Schur-convexity, Schur-geometric and Harmonic convexity of dual form of a class symmetric functions,” Journal of Inequalities and Applications, vol. 2013, p. 295, 2013. View at Publisher · View at Google Scholar
  7. Zh.-H. Yang, “Schur power convexity of Stolarsky means,” Publicationes Mathematicae Debrecen, vol. 80, no. 1-2, pp. 43–66, 2012. View at Google Scholar · View at MathSciNet
  8. Zh.-H. Yang, “Schur power convexity of Gini means,” Bulletin of the Korean Mathematical Society, vol. 50, no. 2, pp. 485–498, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Zh.-H. Yang, “Schur power convexity of the Daróczy means,” Mathematical Inequalities & Applications, vol. 16, no. 3, pp. 751–762, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. W. Wang and S. G. Yang, “Schur m-power convexity of generalized Hamy symmetric function,” Journal of Mathematical Inequalities. In press.
  11. W. Wang and S. G. Yang, “On the Schur m-power convexity for a class of symmetric functions,” Journal of Systems Science and Mathematical Sciences. In press.
  12. B. Y. Wang, Foundations of Majorization Inequalities (in Chinese), Beijing Normal University Press, Beijing, China, 1990.
  13. A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer, New York, NY, USA, 2nd edition, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  14. X. M. Zhang, Geometrically-Convex Functions (in Chinese), Anhui University Press, Hefei, China, 2004.
  15. Y. M. Chu and T. C. Sun, “The Schur harmonic convexity for a class of symmetric functions,” Acta Mathematica Scientia B, vol. 30, no. 5, pp. 1501–1506, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. Wu and F. Qi, “Schur-harmonic convexity for differences of some means,” Analysis, vol. 32, no. 4, pp. 263–270, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. C. P. Niculescu, “Convexity according to the geometric mean,” Mathematical Inequalities & Applications, vol. 3, no. 2, pp. 155–167, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. B. Y. Long and Y. M. Chu, “The Schur convexity and inequalities for a class of symmetric functions,” Acta Mathematica Scientia A, vol. 32, no. 1, pp. 80–89, 2012 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. H. Wu, “Generalization and sharpness of the power means inequality and their applications,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 637–652, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. W. F. Xia and Y. M. Chu, “Schur convexity with respect to a class of symmetric functions and their applications,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 3, pp. 84–96, 2011. View at Google Scholar · View at MathSciNet
  21. H.-T. Ku, M.-C. Ku, and X.-M. Zhang, “Inequalities for symmetric means, symmetric harmonic means, and their applications,” Bulletin of the Australian Mathematical Society, vol. 56, no. 3, pp. 409–420, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. K. Z. Guan, “The Hamy symmetric function and its generalization,” Mathematical Inequalities & Applications, vol. 9, no. 4, pp. 797–805, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. G. H. Hardy, J. E. Littlewood, and G. Pólya, “Some simple inequalities satisfied by convex functions,” Messenger of Mathematics, vol. 58, pp. 145–152, 1929. View at Google Scholar
  24. Y. M. Chu, W. F. Xia, and T. H. Zhao, “Schur convexity for a class of symmetric functions,” Science China. Mathematics, vol. 53, no. 2, pp. 465–474, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. F. Qi, J. Sándor, S. S. Dragomir, and A. Sofo, “Notes on the Schur-convexity of the extended mean values,” Taiwanese Journal of Mathematics, vol. 9, no. 3, pp. 411–420, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, “Generalized convexity and inequalities,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1294–1308, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. D.-M. Li and H.-N. Shi, “Schur convexity and Schur-geometrically concavity of generalized exponent mean,” Journal of Mathematical Inequalities, vol. 3, no. 2, pp. 217–225, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. S. Toader and G. Toader, “Complementaries of Greek means with respect to Gini means,” International Journal of Applied Mathematics & Statistics, vol. 11, no. N07, pp. 187–192, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. W.-F. Xia and Y.-M. Chu, “Schur-convexity for a class of symmetric functions and its applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 493759, 15 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. M. Shaked, J. G. Shanthikumar, and Y. L. Tong, “Parametric Schur convexity and arrangement monotonicity properties of partial sums,” Journal of Multivariate Analysis, vol. 53, no. 2, pp. 293–310, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. H.-N. Shi, Y.-M. Jiang, and W.-D. Jiang, “Schur-convexity and Schur-geometrically concavity of Gini means,” Computers & Mathematics with Applications, vol. 57, no. 2, pp. 266–274, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. A. Forcina and A. Giovagnoli, “Homogeneity indices and Schur-convex functions,” Statistica, vol. 42, no. 4, pp. 529–542, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. Zh.-H. Yang, “Necessary and sufficient condition for Schur convexity of the two-parameter symmetric homogeneous means,” Applied Mathematical Sciences, vol. 5, no. 61–64, pp. 3183–3190, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. Zh.-H. Yang, “Necessary and sufficient conditions for Schur geometrical convexity of the four-parameter homogeneous means,” Abstract and Applied Analysis, vol. 2010, Article ID 830163, 16 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. Zh.-H. Yang, “Schur harmonic convexity of Gini means,” International Mathematical Forum. Journal for Theory and Applications, vol. 6, no. 13–16, pp. 747–762, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. H. N. Shi, Theory of Majorization and Analytic Inequalities, Harbin Institute of Technology Press, Harbin, China, 2013.
  37. Z. Shan, Geometric Inequalities (in Chinese), Shanghai Education Press, Shanghai, China, 1980.
  38. D. S. Mitrinović, J. E. Pečarić, and V. Volenec, Recent Advances in Geometric Inequalities, vol. 28, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989. View at MathSciNet
  39. J. Z. Zhang and L. Yang, “A class of geometric inequalities concerning systems of mass points,” Journal of China University of Science and Technology, vol. 11, no. 2, pp. 1–8, 1981. View at Google Scholar · View at MathSciNet
  40. G. S. Leng, T. Y. Ma, and X. Z. Qian, “Inequalities for a simplex and an interior point,” Geometriae Dedicata, vol. 85, no. 1–3, pp. 1–10, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. S. G. Yang, W. Wang, J. J. Pan, and D. Qian, “The n-dimensional Pedoe inequality in hyperbolic space with applications,” Advances in Mathematics, vol. 41, no. 3, pp. 347–355, 2012. View at Google Scholar · View at MathSciNet
  42. J. Z. Xiao and X. H. Zhu, “Schur convex functions and inequalities of n-simplices,” Applied Mathematics. A Journal of Chinese Universities A, vol. 16, no. 4, pp. 428–434, 2001. View at Google Scholar · View at MathSciNet