BdMO Online Forum

Muirhead's Inequality states that if a sequence $A$ majorizes a sequence $B$, then given a set of positive reals $x_1,x_2 \cdots x_n$:
\[ \sum_{\text{sym}}{x_{1}}^{a_{1}}{x_{2}}^{a_{2}}\cdots{x_{n}}^{a_{n}}\geq\sum_{\text{sym}}{x_{1}}^{b_{1}}{x_{2}}^{b_{2}}\cdots{x_{n}}^{b_{n}} \]

Majorization:
We can say $\{a_i\}$ majorizes $\{b_i\}$ if
\[ \sum_{i=1}^n a_i = \sum_{i=1}^n b_i \]
And \[ \sum_{j=1}^k a_j \leq \sum_{j=1}^k b_j \] for all $k \leq n$