Pythagorean means

In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). They are defined by:

Each mean has the following properties:

These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music. The harmonic and arithmetic means are reciprocal duals of each other for positive arguments (${\displaystyle HM(1/x_{1}\ldots 1/x_{n})=1/AM(x_{1}\ldots x_{n})}$) while the geometric mean is its own reciprocal dual.

There is an ordering to these means (if all of the ${\displaystyle x_{i}}$ are positive)

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