Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1a2, ..., an are real numbers and let denote the kth elementary symmetric function in a1a2, ..., an. Then the elementary symmetric means, given by

satisfy the inequality

If all the numbers ai are nonzero, then equality holds if and only if all the numbers ai are equal. S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean.

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