Maclaurin's inequality

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

Let ${\displaystyle a_{1},a_{2},\ldots ,a_{n}}$ be non-negative real numbers, and for ${\displaystyle k=1,2,\ldots ,n}$, define the averages ${\displaystyle S_{k}}$ as follows:

The numerator of this fraction is the elementary symmetric polynomial of degree ${\displaystyle k}$ in the ${\displaystyle n}$ variables ${\displaystyle a_{1},a_{2},\ldots ,a_{n}}$, that is, the sum of all products of ${\displaystyle k}$ of the numbers ${\displaystyle a_{1},a_{2},\ldots ,a_{n}}$ with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient ${\displaystyle {\tbinom {n}{k}}.}$

Maclaurin's inequality is the following chain of inequalities:

with equality if and only if all the ${\displaystyle a_{i}}$ are equal.

For ${\displaystyle n=2}$, this gives the usual inequality of arithmetic and geometric means of two non-negative numbers. Maclaurin's inequality is well illustrated by the case ${\displaystyle n=4}$:

Maclaurin's inequality can be proved using Newton's inequalities or generalised Bernoulli's inequality.

See also

• Newton's inequalities
• Muirhead's inequality
• Generalized mean inequality
• Bernoulli's inequality

References

• 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.