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 $x^{n-i}$ in $(x+a_1)\dots(x+a_n)$. For example, for $n=3$,

$$\sigma_0 = 1$$

$$\sigma_1 = a_1 + a_2 + a_3$$

$$\sigma_2 = a_1 a_2 + a_2 a_3 + a_3 a_1$$

$$\sigma_3 = a_1 a_2 a_3.$$

The $i$-th elementary symmetric mean $S_i$ is 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,

$$S_0 = 1$$

$$S_1 = \frac{a_1 + a_2 + a_3}{3}$$

$$S_2 = \frac{a_1 a_2 + a_2 a_3 + a_3 a_1}{3}$$

$$S_3 = a_1 a_2 a_3.$$

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

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

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