In mathematics, the geometric–harmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we form the geometric mean of g0 = x and h0 = y and call it g1, i.e. g1 is the square root of xy. We also 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. These may be done sequentially (in any order) or simultaneously.
Now we can iterate this operation with g1 taking the place of x and h1 taking the place of y. In this way, two interdependent sequences (gn) and (hn) are defined:
gn+1=\sqrt{gnhn}
and
hn+1=
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Both of these sequences converge to the same number, which we call the geometric–harmonic mean M(x, y) of x and y. The geometric–harmonic mean is also designated as the harmonic–geometric 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 arithmetic–geometric 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 arithmetic–geometric mean, then we also have
M(x,y)=
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We have the following inequality involving the Pythagorean means and iterated Pythagorean means :
min(x,y)\leqH(x,y)\leqHG(x,y)\leqG(x,y)\leqGA(x,y)\leqA(x,y)\leqmax(x,y)
where the iterated Pythagorean means have been identified with their parts in progressing order: