Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a₁, a₂, ..., a_n are real numbers and let ${\displaystyle \sigma _{k}}$ denote the kth elementary symmetric function in a₁, a₂, ..., a_n. Then the elementary symmetric means, given by

satisfy the inequality

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

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