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>, <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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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>]]
  
 
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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 $x_1,\ldots,x_n$

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

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

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

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