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In mathematics, the geometric-harmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the geometric mean of g_{0} = x and h_{0} = y and call it g_{1}, i.e. g_{1} is the square root of xy. We then form the harmonic mean of x and y and call it h_{1}, i.e. h_{1} is the reciprocal of the arithmetic mean of the reciprocals of x and y.## See also

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Now we can iterate this operation with g_{1} taking the place of x and h_{1} taking the place of y. In this way, two sequences (g_{n}) and (h_{n}) are defined:

- $g\_\{n+1\}\; =\; sqrt\{g\_n\; h\_n\}$

and

- $h\_\{n+1\}\; =\; frac\{2\}\{frac\{1\}\{g\_n\}\; +\; frac\{1\}\{h\_n\}\}$

Both of these sequences converge to the same number, which we call the geometric-harmonic mean M(x, y) of x and y.

M(x, y) is a number between the geometric and harmonic mean of x and y; in particular it is between x and y. If r > 0, then M(rx, ry) = r M(x, y).

If AG(x, y) is the arithmetic-geometric mean, then we also have

- $M(x,y)\; =\; frac\{1\}\{AG(frac\{1\}\{x\},frac\{1\}\{y\})\}$

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Last updated on Friday September 26, 2008 at 06:52:48 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday September 26, 2008 at 06:52:48 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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