Newton's Inequality

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1 Background
2 Statement
3 Proof
4 See Also

Background

For $x_1, \ldots, x_n$ , we define the symmetric sum $s_k$ to be the coefficient of $t^{n-k}$ in the polynomial $\prod_{i=1}^{n}(t+x_i)$ (see Viete's sums). We define the symmetric average $d_k$ to be $\textstyle s_k/{n \choose k}$ .

Statement

For non-negative $x_1, \ldots, x_n$ and $0 < k < n$ ,

$d_k^2 \ge d_{k-1}d_{k+1}$ ,

with equality exactly when all the $x_i$ are equal.

Proof

Lemma. For real $x_1 , \ldots, x_m$ , there exist real $x'_1, \ldots, x'_{m-1}$ with the same symmetric averages $d_0, \ldots, d_{m-1}$ .

Proof. We consider the derivative of $P(t) = \prod_{i=1}^n (t+x_i)$ . The roots of $P$ are $-x_1, \ldots, -x_n$ . Without loss of generality, we assume that the $x_i$ increase as $i$ increases. Now for any $i \in [1, m-1]$ , $P'(t)$ must have a root between $x_i$ and $x_{i+1}$ by Rolle's theorem if $x_i \neq x_{i+1}$ , and if $x_i = x_{i+1} = \cdots = x_{i+k}$ , then $x_{i}$ is a root of $P$ $k+1$ times, so it must be a root of $P'$ $k$ times. It follows that $P'$ must have $m-1$ non-positive, real roots, i.e., for some non-negative reals $x'_1, \ldots, x'_{m-1}$ ,

${}P'(t) = m \prod_{i=1}^{m-1}(t+x'_i)$ .

It follows that the symmetric sum $s'_k$ for $x'_1, \ldots, x'_{m-1}$ is $\frac{m-k}{m}s_k$ , so the symmetric average $d'_k = \frac{s'_k}{{m-1 \choose k}} = \frac{m-k}{m} \cdot \frac{s_k}{{m-1 \choose k}} = \frac{s_k}{{m \choose k}} = d_k$ .