Pythagorean means

The three classical Pythagorean means are the arithmetic mean ("A"), the geometric mean ("G"), and the harmonic mean ("H"). They are defined by:

* $A(x\_1,\; ldots,\; x\_n)\; =\; frac\{1\}\{n\}(x\_1\; +\; cdots\; +\; x\_n)$
* $G(x\_1,\; ldots,\; x\_n)\; =\; sqrt\; [n]\; \{x\_1\; cdots\; x\_n\}$
* $H(x\_1,\; ldots,\; x\_n)\; =\; frac\{n\}\{frac\{1\}\{x\_1\}\; +\; cdots\; +\; frac\{1\}\{x\_n$

Each of these means satisfies the properties:

* $M(x,x,\; ldots,x)\; =\; x$
* $M(bx\_1,\; ldots,\; bx\_n)\; =\; b\; M(x\_1,\; ldots,\; x\_n)$

There is an ordering to these means (if all of the $x\_i$ are positive):

: $A(x\_1,ldots,x\_n)\; geq\; G(x\_1,ldots,x\_n)\; geq\; H(x\_1,ldots,x\_n)$

with equality holding if and only if the $x\_i$ 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.

ee also

* arithmetic-geometric mean
* average

External links

* [http://mathworld.wolfram.com/PythagoreanMeans.html Pythagorean means on MathWorld]