Pythagorean Means

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Pythagorean Means

$A(x_1, \cdots, x_n) = rac{1}{n}(x_1 + \cdots + x_n)$

$G(x_1, \cdots, x_n) = \sqrt$ {Link without Title} {x_1 \cdots x_n}

$H(x_1, \cdots, x_n) = rac{n}{rac{1}{x_1} + \cdots + rac{1}{x_n}}$

Each of these means satisfies the properties:

$M(x,x,\cdots,x) = x$

$M(bx_1, \cdots, bx_n) = b M(x_1, \cdots, x_n)$

There is an ordering to these means:

:

A(x_1,\cdots,x_n) \geq G(x_1,\cdots,x_n) \geq H(x_1,\cdots,x_n)

with equality holding if and only if the

x_i

are all equal. This is a generalization of the Arithmetic Mean-Geometric Mean Inequality and a special case of an inequality for Generalized Mean s.

SEE ALSO

Arithmetic Mean

Arithmetic-geometric Mean

Average

Generalized Mean

Harmonic Mean

Inequality Of Arithmetic And Geometric Means

EXTERNAL LINKS

Pythagorean Means on MathWorld