In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means.
Let
,
a
2
, … ,
a
n
{\displaystyle a_{1},a_{2},\ldots ,a_{n}}
be non-negative real numbers, and for
{\displaystyle k=1,2,\ldots ,n}
, define the averages
as follows:
=
k
a
The numerator of this fraction is the elementary symmetric polynomial of degree
variables
, that is, the sum of all products of
of the numbers
with the indices in increasing order.
The denominator is the number of terms in the numerator, the binomial coefficient
Maclaurin's inequality is the following chain of inequalities:
with equality if and only if all the
are equal.
, 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 {\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 Newton's inequalities or generalised Bernoulli's inequality.
This article incorporates material from MacLaurin's Inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.