Maclaurin's Inequality

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

Let $a_1, a_2,\ldots,a_n$ be non-negative real numbers, and for $k=1,2,\ldots,n$, define the averages $S_k$ as follows:

$S_k = \frac.$

The numerator of this fraction is the elementary symmetric polynomial of degree $k$ in the $n$ variables $a_1, a_2,\ldots,a_n$, that is, the sum of all products of $k$ of the numbers $a_1, a_2,\ldots,a_n$ with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient $\tbinom n k.$ Maclaurin's inequality is the following chain of inequalities:

S_1 \geq \sqrt \geq \sqrt \geq \cdots \geq \sqrt /math>,

with equality if and only if all the $a_i$ are equal.

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

\begin
&\quad \frac \\ pt&\ge \sqrt \\ pt&\ge \sqrt \\ pt&\ge \sqrt
\end

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

See also

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

References

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{{PlanetMath attribution, id=3835, title=MacLaurin's Inequality

Real analysis
Inequalities (mathematics)
Symmetric functions