MacLaurin's inequality
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In mathematics, MacLaurin's inequality, named after Colin Maclaurin, is a generalization, or more specifically, a refinement, of the inequality of arithmetic and geometric means.
Let 
be positive real numbers, and for 
define the averages Sk as follows:
The numerator of this fraction is the elementary symmetric polynomial of degree k in the n variables , that is, the sum of all products of k 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 states that then 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}](../../../math/f/f/e/ffe8b7a45d0608396daf07402fe919df.png)
with equality if and only if all the ai's are equal.
For n = 2 this gives the usual inequality of arithmetic and geometric means of two numbers. For n = 3 MacLaurin's inequality becomes
![\frac{a_1+a_2+a_3}{3} \ge \sqrt{\frac{a_1a_2+a_1a_3+a_2a_3}{3}} \ge \sqrt[3]{a_1a_2a_3}.](../../../math/d/7/f/d7fdde0d5a2125e616b41909358a80b6.png)
The Maclaurin inequalities can be proved using the Newton's inequalities.
[edit] See also
[edit] References
- Biler, Piotr; Witkowski, Alfred (1990). Problems in mathematical analysis. New York, N.Y.: M. Dekker. ISBN 0824783123.
[edit] External links
This article incorporates material from MacLaurin's Inequality on PlanetMath, which is licensed under the GFDL.
