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{displaystyle sum_{ 1leq i_1 < cdots < i_k leq n}a_{i_1} a_{i_2} cdots a_{i_k{displaystyle {n choose k.

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 {nchoose k}.

MacLaurin's inequality states that the following chain of inequalities is true:

:$$S_1 geq sqrt{S_2} geq sqrt [3] {S_3} geq cdots geq sqrt [n] {S_n}

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:

: $$egin{align}& {} quad frac{a_1+a_2+a_3+a_4}{4} \ \& {} ge sqrt{frac{a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4}{6 \ \& {} ge sqrt [3] {frac{a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4}{4 \ \& {} ge sqrt [4] {a_1a_2a_3a_4}.end{align}

Maclaurin's inequality can be proved using the Newton's inequalities.

ee also

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

References

*cite book
last = Biler
first = Piotr
coauthors = Witkowski, Alfred
title = Problems in mathematical analysis
publisher = New York, N.Y.: M. Dekker
date = 1990
pages =
isbn = 0824783123

External links

* [http://mcraefamily.com/MathHelp/BasicNumberIneq.htm Famous Inequalities]

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