What is AM-GM-HM inequality

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

$${\rm AM} \geq {\rm GM} \geq {\rm HM}\;.$$

Extended explanation

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

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

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

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

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

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