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What is AM-GM-HM inequality

  1. Jul 23, 2014 #1
    Definition/Summary

    Arithmetic-Geometric-Harmonic Means Inequality is a relationship between the size of the arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM) of a given set of positive real numbers. It is often very useful in analysis.

    Equations

    [tex]{\rm AM} \geq {\rm GM} \geq {\rm HM}\;.[/tex]

    Extended explanation

    The arithmetic mean is given by
    [tex]A(n,a_i) = \frac{1}{n}\sum_{i=1}^{n} a_i = \frac{a_1 + a_2+\cdots+a_n}{n}\;.[/tex]

    The geometic mean is given by
    [tex]G(n,a_i) = \left(\prod_{i=1}^{n} a_i\right)^{1/n}
    =\left(a_1 a_2\cdots a_n\right)^{1/n}\;.[/tex]

    The harmonic mean is given by
    [tex]H(n,a_i) = \frac{n}{\sum_{i=1}^{n} \frac{1}{a_i}} = \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+ \cdots+\frac{1}{a_n}}
    \;.[/tex]

    The statement of the AM-GM-HM inequality is:

    For [tex]x_i > 0, i = 1,2,\cdots, k\, ,[/tex]
    we have [tex]A(k,x_i) \geq G(k,x_i) \geq H(k,x_i)\;.[/tex]
    equality holds if and only if [tex]x_i=x_j\;, \forall\; i, j[/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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