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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 x_1,\ldots,x_n 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+\fr...

with equality if and only if x_1=x_2=\cdots=x_n. This inequality can be expanded to the power mean inequality.

The inequality is clearly shown in this diagram for n=2

As a consequence we can have the following inequality: If x_1,x_2,\cdots,x_n are positive reals, then (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 with equality if and only if x_1=x_2=\cdots=x_n; which follows directly by cross multiplication from the AM-HM inequality.This is extremely useful in problem solving.



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