Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
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The Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality (RMS-AM-GM-HM), is an inequality of the root-mean square, arithmetic mean, geometric mean, and harmonic mean of a set of positive real numbers 
that says:
![\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://alt2.artofproblemsolving.com/Forum/latexrender/pictures/c/5/5/c55513a48589cd9f3e5cb2fd459db454ecb9c738.gif "Click on the formula to view the LaTeX code")
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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