mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Maclaurin's inequality, named after

Colin Maclaurin Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for bei ...

, is a refinement of the

inequality of arithmetic and geometric means In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...

. Let ''a''₁, ''a''₂, ..., ''a''_''n'' be

positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s, 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 In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sym ...

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 In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

\scriptstyle

.

Maclaurin's inequality is the following chain of

inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

: :

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

with equality

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

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: :

\end

Maclaurin's inequality can be proved using

Newton's inequalities In mathematics, the Newton inequalities are named after Isaac Newton. Suppose ''a''1, ''a''2, ..., ''a'n'' are real numbers and let e_k denote the ''k''th elementary symmetric polynomial in ''a''1, ''a''2, ..., ''a ...

References

* {{PlanetMath attribution, id=3835, title=MacLaurin's Inequality Real analysis Inequalities Symmetric functions

See also

References