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:

  • 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 xi are positive):

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

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.

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