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In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1, a2, ..., an are real numbers and let e k {\displaystyle e_{k}} denote the kth elementary symmetric polynomial in a1, a2, ..., an. Then the elementary symmetric means, given by
satisfy the inequality
If all the numbers ai are non-zero, then equality holds if and only if all the numbers ai are equal.
It can be seen that S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean.
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