Pythagorean means
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The three classical Pythagorean means are the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H). They are defined by:
![G(x_1, \cdots, x_n) = \sqrt[n]{x_1 \cdots x_n}](../../../math/f/6/6/f6627e3f551da370b0709f0447ef2031.png)
Each of these means satisfies the properties:
There is an ordering to these means (if all of the xi are positive):
with equality holding if and only if the xi are all equal. This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means.
