Symmetric function inequalities

Given numbers $a_1,\dots,a_n$ and $0 \le i \le n$ , the i-th elementary symmetric function $\sigma_i$ is defined to be the coefficient of x^n-iin $(x+a_1)\dots(x+a_n)$ . For example, for n=3,
$\begin{align*}\sigma_0 &= 1 \\ \sigma_1 &= a_1 + a_2 + a_3 \\ \sigma_2 &= a_1 a_2 + a_2 a_3 + a_3 a_1 \\ \sigma_3 &= a_1 a_2 a_3. \end{align*}$
The i-th elementary symmetric mean S_iis the arithmetic mean of the monomials appearing in the expansion of $\sigma_i$ ; in other words, $S_i:=\sigma_i/\binom{n}{i}$ . In the example above,
$\begin{align*}S_0 &= 1 \\ S_1 &= \frac{a_1 + a_2 + a_3}{3} \\ S_2 &= \frac{a_1 a_2 + a_2 a_3 + a_3 a_1}{3} \\ S_3 &= a_1 a_2 a_3. \end{align*}$

Newton's inequality:
For any real numbers $a_1,\dots,a_n$ , we have $S_{i-1} S_{i+1} \le S_i^2$ .

Maclaurin's inequality:
For $a_1,\dots,a_n \ge 0$ , we have

$\begin{displaymath}S_1 \ge \sqrt{S_2} \ge \sqrt[3]{S_3} \ge \dots \ge \sqrt[n]{S_n}.\end{displaymath}$

Moreover, if the a_i are positive and not all equal, then the inequalities are all strict.