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In mathematics, the geometric–harmonic mean M 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 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 and are defined:
and
Both of these sequences converge to the same number, which we call the geometric–harmonic mean M of x and y. The geometric–harmonic mean is also designated as the harmonic–geometric mean.