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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 number]]s <math>x_1,\ldots,x_n</math> that says:

<math>\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}}</math>

with equality if and only if <math>x_1=x_2=\cdots=x_n</math>.  This inequality can be expanded to the [[power mean inequality]].

[[Image:RMS-AM-GM-HM.gif|frame|right|The inequality is clearly shown in this diagram for <math>n=2</math>]]

As a consequence we can have the following inequality:
If <math>x_1,x_2,\cdots,x_n</math> are positive reals, then 
<math>(x_1+x_2+\cdots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\cdots \frac{1}{x_n}\right) \geq n^2</math> with equality if and only if <math>x_1=x_2=\cdots=x_n</math>; which follows directly by cross multiplication from the AM-HM inequality.This is extremely useful in problem solving.




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[[Category:Inequality]]
[[Category:Theorems]]