# Muirhead’s theorem

Let $0\leq s_{1}\leq\cdots\leq s_{n}$ and $0\leq t_{1}\leq\dots\leq t_{n}$ be real numbers such that

 $\sum_{i=1}^{n}s_{i}=\sum_{i=1}^{n}t_{i}\;\mbox{and}\;\;\sum_{i=1}^{k}s_{i}\leq% \sum_{i=1}^{k}t_{i}\;\;(k=1,\dots,n-1)$

Then for any nonnegative numbers $x_{1},\dots,x_{n}$,

 $\sum_{\sigma}x_{1}^{s_{\sigma(1)}}\dots x_{n}^{s_{\sigma(n)}}\geq\sum_{\sigma}% x_{1}^{t_{\sigma(1)}}\dots x_{n}^{t_{\sigma(n)}}$

where the sums run over all permutations $\sigma$ of $\{1,2,\dots,n\}$.

Title Muirhead’s theorem MuirheadsTheorem 2013-03-22 13:15:25 2013-03-22 13:15:25 Koro (127) Koro (127) 7 Koro (127) Theorem msc 26D15 Muirhead’s inequality CharacterizationsOfMajorization