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, ..., ''a''_''n'' be positive real numbers, and for ''k'' = 1, 2, ..., ''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, ..., ''a''_''n'', that is, the sum of all products of ''k'' of the numbers ''a''₁, ''a''₂, ..., ''a''_''n'' with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient $\scriptstyle$$.$

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 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.

See also

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

References

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

Real analysis
Inequalities
Symmetric functions