Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality

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 $x_1,\ldots,x_n$

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

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 $n=2$

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