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 (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

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

See also

• arithmetic-geometric mean
• arithmetic-harmonic mean
• mean

External links

• Harmonic-Geometric Mean on Mathworld