Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 67624, 9 pages

Schur-convexity of the complete elementary symmetric function

Kaizhong Guan

Department of Mathematics and Physics, Nanhua University, Hengyang, Hunan 421001, China

Received 2 October 2004; Revised 15 January 2005; Accepted 27 January 2005

Copyright © 2006 Kaizhong Guan. 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.


We prove that the complete elementary symmetric function cr=cr(x)=Cn[r](x)=i1++in=rx1i1xnin and the function φr(x)=cr(x)/cr1(x) are Schur-convex functions in R+n={(x1,x2,,xn)|xi>0}, where i1,i2,,in are nonnegative integers, rN={1,2,}, i=1,2,,n. For which, some inequalities are established by use of the theory of majorization. A problem given by K. V. Menon (Duke Mathematical Journal 35 (1968), 37–45) is also solved.