IJPAM: Volume 58, No. 4 (2010)

SCHUR CONVEXITY AND INEQUALITIES FOR
A CLASS OF SYMMETRIC FUNCTIONS

Weifeng Xia$^1$, Gendi Wang$^2$, Yuming Chu$^3$
$^1$School of Teachers' Education
Huzhou Teachers College
Huzhou, 313000, P.R. CHINA
e-mail: xwf212@hutc.zj.cn
$^{2,3}$Department of Mathematics
Huzhou Teachers College
Huzhou, 313000, P.R. CHINA
$^2$e-mail: wgdi@hutc.zj.cn
$^3$e-mail: chuyuming@hutc.zj.cn


Abstract.For $x=(x_{1},x_{2},\cdots,x_{n})\in R_{+}^{n}$, the symmetric function $F_{n}(x,r)$ is defined by

\begin{displaymath}F_{n}(x,r)=F_{n}(x_{1},x_{2},\cdots,x_{n};r)=\sum_{1\leq i_{1... ...dots< i_{r}\leq n}\prod_{j=1}^{r}\frac{x_{i_{j}}}{1+x_{i_{j}}},\end{displaymath}

where $r=1,2,\cdots,n$ and $i_{1},i_{2},\cdots,i_{n}$ are positive integers.

In this article, the Schur convexity, Schur harmonic convexity and Schur multiplicative convexity of $F_{n}(x,r)$ are discussed. As applications, some inequalities are established by use of the theory of majorization.

Received: January 8, 2010

AMS Subject Classification: 05E05

Key Words and Phrases: symmetric function, Schur convex, Schur multiplicatively convex, Schur harmonic convex, theory of majorization

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 58
Issue: 4