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Newton's inequalities

From Wikipedia, the free encyclopedia

In mathematics, the Newton inequalities are named after Isaac Newton. 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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  • Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities. Cambridge University Press. ISBN 978-0521358804.