Muirhead’s theorem

Let $0\le s_1\le \mathrm{\cdots }\le s_n$ and $0\le t_1\le \mathrm{\dots }\le t_n$be real numbers such that

$$\sum _{i=1}^n{s}_i=\sum _{i=1}^n{t}_i\text{and}\sum _{i=1}^k{s}_i\le \sum _{i=1}^k{t}_i(k=1,\mathrm{\dots },n-1)$$

Then for any nonnegative numbers $x_1,\mathrm{\dots },x_n$,

$$\sum _{\sigma }{x}_1^{s_{\sigma (1)}}\mathrm{\dots }{x}_n^{s_{\sigma (n)}}\ge \sum _{\sigma }{x}_1^{t_{\sigma (1)}}\mathrm{\dots }{x}_n^{t_{\sigma (n)}}$$

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