, the geometric-harmonic mean
) of two positive real numbers x
is defined as follows: we first form the geometric mean
and call it g1
, i.e. g1
is the square root
. We then form the harmonic mean
and call it h1
, i.e. h1
is the reciprocal
of the arithmetic mean
of the reciprocals of x
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:
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