Maclaurin's inequality

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

Let a₁, a₂, ..., a_n be positive real numbers, and for k = 1, 2, ..., n define the averages S_k as follows:

$S_{k}={\frac {\displaystyle \sum _{{1\leq i_{1}<\cdots <i_{k}\leq n}}a_{{i_{1}}}a_{{i_{2}}}\cdots a_{{i_{k}}}}{\displaystyle {n \choose k}}}.$

The numerator of this fraction is the elementary symmetric polynomial of degree k in the n variables a₁, a₂, ..., a_n, that is, the sum of all products of k of the numbers a₁, a₂, ..., a_n with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient $\scriptstyle {n \choose k}.$

Maclaurin's inequality is the following chain of inequalities:

$S_{1}\geq {\sqrt {S_{2}}}\geq {\sqrt[ {3}]{S_{3}}}\geq \cdots \geq {\sqrt[ {n}]{S_{n}}}$

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:

${\begin{aligned}&{}\quad {\frac {a_{1}+a_{2}+a_{3}+a_{4}}{4}}\\[8pt]&{}\geq {\sqrt {{\frac {a_{1}a_{2}+a_{1}a_{3}+a_{1}a_{4}+a_{2}a_{3}+a_{2}a_{4}+a_{3}a_{4}}{6}}}}\\[8pt]&{}\geq {\sqrt[ {3}]{{\frac {a_{1}a_{2}a_{3}+a_{1}a_{2}a_{4}+a_{1}a_{3}a_{4}+a_{2}a_{3}a_{4}}{4}}}}\\[8pt]&{}\geq {\sqrt[ {4}]{a_{1}a_{2}a_{3}a_{4}}}.\end{aligned}}$

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

See also

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