Definitions

# Geometric

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 g0 = x and h0 = y and call it g1, i.e. g1 is the square root of xy. We then form the harmonic mean of x and y and call it h1, i.e. h1 is the reciprocal of the arithmetic mean of the reciprocals of x and y.

Now we can iterate this operation with g1 taking the place of x and h1 taking the place of y. In this way, two sequences (gn) and (hn) are defined:

$g_\left\{n+1\right\} = sqrt\left\{g_n h_n\right\}$

and

$h_\left\{n+1\right\} = frac\left\{2\right\}\left\{frac\left\{1\right\}\left\{g_n\right\} + frac\left\{1\right\}\left\{h_n\right\}\right\}$

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

$M\left(x,y\right) = frac\left\{1\right\}\left\{AG\left(frac\left\{1\right\}\left\{x\right\},frac\left\{1\right\}\left\{y\right\}\right)\right\}$