Newton's inequalities

In mathematics, the Newton inequalities refer to a set of mathematical inequalities related to mathematical series. These inequalities are named after Isaac Newton who proved the theorem in 1707.[1] Suppose a1a2, ..., an are non-negative real numbers and let denote the kth elementary symmetric polynomial in a1a2, ..., an. Then the elementary symmetric means, given by

satisfy the inequality

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.

See also

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References

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  1. ^ Newton, Isaac (1707). Arithmetica universalis: sive de compositione et resolutione arithmetica liber.

Other

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