Difference between revisions of "Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality"
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Revision as of 15:10, 26 October 2007
The RMS-AM-GM-HM inequality, which stands for root-mean square-arithmetic mean-geometric mean-harmonic mean, says that for any positive real numbers 
![$\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}\ge\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}$](http://latex.artofproblemsolving.com/c/e/b/ceb0cbb7b637caa6f14856d9350586808892e419.png)
with equality if and only if 
. This inequality can be expanded to the power mean inequality.
The inequality is clearly shown in this diagram for 
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