In mathematics, the geometricâÂÂharmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we form the geometric mean of g<sub>0</sub> = x and h<sub>0</sub> = y and call it g<sub>1</sub>, i.e. g<sub>1</sub> is the square root of xy. We also form the harmonic mean of x and y and call it h<sub>1</sub>, i.e. h<sub>1</sub> is the reciprocal of the arithmetic mean of the reciprocals of x and y. These may be done sequentially (in any order) or simultaneously.
Now we can iterate this operation with g<sub>1</sub> taking the place of x and h<sub>1</sub> taking the place of y. In this way, two interdependent sequences (g<sub>n</sub>) and (h<sub>n</sub>) are defined:
and
Both of these sequences converge to the same number, which we call the geometricâÂÂharmonic mean M(x, y) of x and y. The geometricâÂÂharmonic mean is also designated as the harmonicâÂÂgeometric mean. (cf. Wolfram MathWorld below.)
The existence of the limit can be proved by the means of Bolzano–Weierstrass theorem in a manner almost identical to the proof of existence of arithmeticâÂÂgeometric mean.
M(x, y) is a number between the geometric and harmonic mean of x and y; in particular it is between x and y. M(x, y) is also homogeneous, i.e. if r > 0, then M(rx, ry) = r M(x, y).
If AG(x, y) is the arithmeticâÂÂgeometric mean, then we also have
We have the following inequality involving the Pythagorean means {H, G, A} and iterated Pythagorean means {HG, HA, GA}:
where the iterated Pythagorean means have been identified with their parts {H, G, A} in progressing order: