Maclaurin's inequality - Alchetron, the free social encyclopedia

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 = ∑ 1 ≤ i 1 < ⋯ < i k ≤ n a i 1 a i 2 ⋯ a i k ( n 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 ( n k ) .

Maclaurin's inequality is the following chain of inequalities:

S 1 ≥ S 2 ≥ S 3 3 ≥ ⋯ ≥ S n 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:

a 1 + a 2 + a 3 + a 4 4 ≥ 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 ≥ 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 3 ≥ a 1 a 2 a 3 a 4 4 .

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

References

Maclaurin's inequality Wikipedia

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