×

zbMATH — the first resource for mathematics

Schur-convexity of the complete elementary symmetric function. (English) Zbl 1107.05093
The main purpose of this paper is to prove that the functions \(c_{r}(x)=\sum _{i_1+\cdots +i_{n}=r}x_{1}^{i_{1}}\dots x_{n}^{i_{n}}\) and \(c_{r}(x)/c_{r-1}(x)\), where \(i_{1},\dots ,i_{n}\) are non-negative integers and \(r\geq 1\), are Schur-convex in \(R_{+}^{n}\) and increasing in \(x_{i}\) for \(i=1,\dots ,n\). Also some inequalities, including the Ky Fan type inequality, are established by use of the theory of majorization.

MSC:
05E05 Symmetric functions and generalizations
26E60 Means
26D20 Other analytical inequalities
PDF BibTeX XML Cite
Full Text: DOI EuDML