# Difference between revisions of "Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality"

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− | The '''RMS-AM-GM-HM''' inequality, which stands for [[root-mean square]]-[[arithmetic mean]]-[[geometric mean]]-[[harmonic mean]], says that for any positive numbers <math>x_1,\ldots,x_n</math> | + | The '''RMS-AM-GM-HM''' inequality, which stands for [[root-mean square]]-[[arithmetic mean]]-[[geometric mean]]-[[harmonic mean]], says that for any positive numbers <math>x_1,\ldots,x_n</math> |

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+ | <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> | ||

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

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

{{stub}} | {{stub}} |

## Revision as of 21:42, 26 June 2006

The **RMS-AM-GM-HM** inequality, which stands for root-mean square-arithmetic mean-geometric mean-harmonic mean, says that for any positive numbers

with equality if and only if . This inequality can be expanded to the generalized mean inequality.

*This article is a stub. Help us out by expanding it.*