Maclaurin's inequality

In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means.

Let be non-negative real numbers, and for , define the averages as follows: The numerator of this fraction is the elementary symmetric polynomial of degree in the variables , that is, the sum of all products of of the numbers with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient

Maclaurin's inequality is the following chain of inequalities: with equality if and only if all the are equal.

For , this gives the usual inequality of arithmetic and geometric means of two non-negative numbers. Maclaurin's inequality is well illustrated by the case : Maclaurin's inequality can be proved using Newton's inequalities or generalised Bernoulli's inequality.

See also

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References

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  • Biler, Piotr; Witkowski, Alfred (1990). Problems in mathematical analysis. New York, N.Y.: M. Dekker. ISBN 0-8247-8312-3.

This article incorporates material from MacLaurin's Inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.