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Maclaurin's Inequality

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Maclaurin's Inequality is an inequality in symmetric polynomials. For notation and background, we refer to Newton's Inequality.

Statement

For non-negative x_1, \ldots, x_n,

d_1 \ge d_2^{1/2} \ge \ldots \ge d_n^{1/n},

with equality exactly when all the x_i are equal.

Proof

By the lemma from Newton's Inequality, it suffices to show that for any n,

d_{n-1}^{1/(n-1)} \ge d_{n}^{1/n}.

Since this is a homogenous inequality, we may normalize so that d_n = \prod x_i = 1. We then transform the inequality to

\frac{\sum 1/x_i}{n} \ge 1^{\frac{n-1}{n}} = 1.

Since the geometric mean of 1/x_1, \ldots, 1/x_n is 1, the inequality is true by AM-GM.

Resources

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