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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]
* 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!
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!