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:

$\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}\ge\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}$

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 $(x_1+x_2+\cdots+x_n)(\frac{1}{x_1}+\frac{1}{x_2}+\cdots \frac{1}{x_n}) \geq n^2$ 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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Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality

From AoPSWiki