Maclaurin's Inequality is an inequality in symmetric polynomials. For notation and background, we refer to Newton's Inequality.
For non-negative ,
with equality exactly when all the are equal.
By the lemma from Newton's Inequality, it suffices to show that for any ,
Since this is a homogenous inequality, we may normalize so that . We then transform the inequality to
Since the geometric mean of is 1, the inequality is true by AM-GM.