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**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]

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