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Muirhead's Inequality states that if a sequence $A$ majorizes a sequence $B$, then given a set of positive reals $x_1,x_2,\cdots,x_n$: $$\sum_{\text{sym}} {x_1}^{a_1}{x_2}^{a_2}\cdots {x_n}^{a_n}\geq \sum_{\text{sym}} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}$$

## Example

The inequality is easier to understand given an example. Since the sequence $(5,1)$ majorizes $(4,2)$ (as $5>4, 5+1=4+2$), Muirhead's inequality states that for any positive $x,y$,

$x^5y^1+y^5x^1\geq x^4y^2+y^4x^2$.

## Proof

We will first prove an important fact. If we have a sequence $C$ such that $\sum_{i=1}^{n} c_i = 0$, then the following result holds: $$\sum_{\text{sym}}{x_1}^{c_1}{x_2}^{c_2}\cdots {x_n}^{c_n}\geq n!$$ for any positive reals $x_1, x_2, ..., x_n$.

Proof: By AM-GM, we know: $$\frac{\sum_{\text{sym}}{x_1}^{c_1}{x_2}^{c_2}\cdots {x_n}^{c_n}}{n!}\geq \sqrt[n!]{\prod_{\text{sym}}{x_1}^{c_1}{x_2}^{c_2}\cdots {x_n}^{c_n}}$$

However, expanding the right hand side, we see $$\sqrt[n!]{\prod_{\text{sym}}{x_1}^{c_1}{x_2}^{c_2}\cdots {x_n}^{cn}}=\sqrt[n!]{\prod_{i=1}^{n} x_i^{(n-1)!\left(c_1+c_2+\cdots +c_n\right)}}=1$$ or $$\frac{\sum_{\text{sym}}{x_1}^{c_1}{x_2}^{c_2}\cdots {x_n}^{c_n}}{n!}\geq 1 \Rightarrow \sum_{\text{sym}}{x_1}^{c_1}{x_2}^{c_2}\cdots {x_n}^{c_n}\geq n!$$

We define our sequence $C$ such that $$c_i=a_i-b_i \hspace{1cm} \forall \hspace{2mm}1\leq i\leq n$$

Note that $$\sum_{i=1}^{n} c_i = \sum_{i=1}^{n} a_i-b_i=\sum_{i=1}^{n} a_i - \sum_{i=1}^{n} b_i=0$$

Therefore, we get that $$\sum_{\text{sym}}\left({x_1}^{c_1}{x_2}^{c_2}\cdots {x_n}^{c_n}\right) -n!\geq 0$$

We multiply this by $\sum_{\text{sym}} \prod_{i=1}^{n}x_i^{b_i}$ to get: $$\left(\sum_{\text{sym}} \prod_{i=1}^{n}x_i^{b_i}\right)\left(\sum_{\text{sym}}\left({x_1}^{c_1}{x_2}^{c_2}\cdots {x_n}^{c_n}-1\right)\right)=\sum_{\text{sym}} \prod_{i=1}^{n} x_i^{b_i+c_i}-\prod_{i=1}^{n} x_i^{b_i}\geq 0$$

Since $b_i+c_i=b_i+a_i-b_i=a_i$, we get $$\sum_{\text{sym}} \prod_{i=1}^{n} x_i^{a_i}-\prod_{i=1}^{n} x_i^{b_i}\geq 0$$ or $$\sum_{\text{sym}} {x_1}^{a_1}{x_2}^{a_2}\cdots {x_n}^{a_n}\geq \sum_{\text{sym}} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}$$

A common bruteforce technique with inequalities is to clear denominators, multiply everything out, and apply Muirhead's or Schur's. However, it is worth noting that any inequality that can be proved directly with Muirhead can also be proved using the Arithmetic Mean-Geometric Mean inequality. In fact, IMO gold medalist Thomas Mildorf says it is unwise to use Muirhead in an Olympiad solution; one should use an application of AM-GM instead. Thus, it is suggested that Muirhead be used only to verify that an inequality can be proved with AM-GM before demonstrating the full AM-GM proof.

As an example, the above inequality can be proved using AM-GM as follows:

$\frac{x^4+x^4+x^4+y^4}{4}\geq\sqrt[4]{x^{12}y^4}=x^3y$

$\frac{x^4+y^4+y^4+y^4}{4}\geq\sqrt[4]{x^4y^{12}}=xy^3$

$x^4+y^4\geq x^3y+xy^3$.
Multiplying both sides by $xy$ (as both $x$ and $y$ are positive),
$x^5y+xy^5\geq x^4y^2+x^2y^4$