Ukrainian Mathematical Journal

, Volume 61, Issue 10, pp 1541–1555

# Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

• Wei-Feng Xia
• Yu-Ming Chu
Article
For x = (x 1, x 2, …, x n ) ∈ (0, 1 ] n and r ∈ { 1, 2, … , n}, a symmetric function F n (x, r) is defined by the relation
$${F_n}\left( {x,r} \right) = {F_n}\left( {{x_1},{x_2}, \ldots, {x_n};r} \right) = \sum\limits_{1{ \leqslant_1} < {i_2} \ldots {i_r} \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 - {x_{{i_j}}}}}{{{x_{{i_j}}}}}} },$$
where i 1, i 2 ,… , i n are positive integers. This paper deals with the Schur convexity and Schur multiplicative convexity of F n (x, r). As applications, some inequalities are established by using the theory of majorization.

## Keywords

Symmetric Function Isoperimetric Inequality Mathematical Induction Analytic Inequality Linear Experiment
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
H. Alzer, “A short proof of Ky Fan’s inequality,” Arch. Math. (Brno), 27B, 199–200 (1991).
2. 2.
E. F. Beckenbach and R. Bellman, Inequalities, Springer, Berlin (1961).Google Scholar
3. 3.
P. S. Bullen, Handbook of Means and Their Inequalities, Kluwer, Dordrecht (2003).
4. 4.
Y. M. Chu and X. M. Zhang, “Necessary and sufficient conditions such that extended mean values are Schur-convex or Schurconcave,” J. Math. Kyoto Univ., 48, No. 1, 229–238 (2008).
5. 5.
Y. M. Chu, X. M. Zhang, and G. D. Wang, “The Schur geometrical convexity of the extended mean values,” J. Convex Anal., 15, No. 4, 707–718 (2008).
6. 6.
G. M. Constantine, “Schur-convex functions on the spectra of graphs,” Discrete Math., 45, No. 2–3, 181–188 (1985).
7. 7.
K. Z. Guan, “The Hamy symmetric function and its generalization,” Math. Inequal. Appl., 9, No. 4, 797–805 (2006).
8. 8.
K. Z. Guan, “Schur-convexity of the complete symmetric function,” Math. Inequal. Appl., 9, No. 4, 567–576 (2006).
9. 9.
K. Z. Guan, “Some properties of a class of symmetric functions,” J. Math. Anal. Appl., 336, No. 1, 70–80 (2007).
10. 10.
K. Z. Guan, “A class of symmetric functions for multiplicatively convex function,” Math. Inequal. Appl., 10, No. 4, 745–753 (2007).
11. 11.
K. Z. Guan and J. H. Shen, “Schur-convexity for a class of symmetric functions and its applications,” Math. Inequal. Appl., 9, No. 2, 199–210 (2006).
12. 12.
G. H. Hardy, J. E. Littlewood, and G. Pólya, “Some simple inequalities satisfied by convex functions,” Messenger Math., 58, 145–152 (1929).Google Scholar
13. 13.
F. K. Hwang and U. G. Rothblum, “Partition-optimization with Schur convex sum objective functions,” SIAM J. Discrete Math., 18, No. 3, 512–524 (2004/2005).
14. 14.
F. K. Hwang, U. G. Rothblum, and L. Shepp, “Monotone optimal multipartitions using Schur convexity with respect to partial orders,” SIAM J. Discrete Math., 6, No. 4, 533–574 (1993).
15. 15.
W. D. Jiang, “Some properties of dual form of the Hamy’s symmetric function,” J. Math. Inequal., 1, No. 1, 117–125 (2007).
16. 16.
A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York (1979).
17. 17.
M. T. McGregor, “On some inequalities of Ky Fan and Wang-Wang,” J. Math. Anal. Appl., 180, No. 1, 182–188 (1993).
18. 18.
A. M. Mercer, “A short proof of Ky Fan’s arithmetic-geometric inequality,” J. Math. Anal. Appl., 204, No. 3, 940–942 (1996).
19. 19.
M. Merkle, “Convexity, Schur-convexity and bounds for the gamma function involving the digamma function,” Rocky Mountain J. Math., 28, No. 3, 1053–1066 (1998).
20. 20.
D. S. Mitrinovic, Analytic Inequalities, Springer, New York (1970).
21. 21.
C. P. Niculescu, “Convexity according to the geometric mean,” Math. Inequal. Appl., 3, No. 2, 155–167 (2000).
22. 22.
J. Pečarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, New York (1992).
23. 23.
F. Qi, “A note on Schur-convexity of extended mean values,” Rocky Mountain J. Math., 35, No. 5, 1787–1793 (2005).
24. 24.
F. Qi, J. Sándor, S. S. Dragomir, and A. Sofo, “Note on the Schur-convexity of the extended mean values,” Taiwan. J. Math., 9, No. 3, 411–420 (2005).
25. 25.
M. Shaked, J. G. Shanthikumar, and Y. L. Tong, “Parametric Schur convexity and arrangement monotonicity properties of partial sums,” J. Multivar. Anal., 53, No. 2, 293–310 (1995).
26. 26.
H. N. Shi, “Schur-convex functions related to Hadamard-type inequalities,” J. Math. Inequal., 1, No. 1, 127–136 (2007).
27. 27.
H. N. Shi, S. H. Wu, and F. Qi, “An alternative note on the Schur-convexity of the extended mean values,” Math. Inequal. Appl., 9, No. 2, 219–224 (2006).
28. 28.
C. Stepniak, “An effective characterization of Schur-convex functions with applications,” J. Convex Anal., 14, No. 1, 103–108 (2007).
29. 29.
C. Stepniak, “Stochastic ordering and Schur-convex functions in comparison of linear experiments,” Metrika, 36, No. 5, 291–298 (1989).
30. 30.
S. H. Wu, “Generalization and sharpness of the power means inequality and their applications,” J. Math. Anal. Appl., 312, No. 2, 637–652 (2005).
31. 31.
X. M. Zhang, “Optimization of Schur-convex functions,” Math. Inequal. Appl., 1, No. 3, 319–330 (1998).
32. 32.
X. M. Zhang, “Schur-convex functions and isoperimetric inequalities,” Proc. Amer. Math. Soc., 126, No. 2, 461–470 (1998).