Geometric–harmonic mean


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.
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.

Properties

M is a number between the geometric and harmonic mean of x and y; in particular it is between x and y. M is also homogeneous, i.e. if r > 0, then M = r M.
If AG is the arithmetic–geometric mean, then we also have

Inequalities

We have the following inequality involving the Pythagorean means and iterated Pythagorean means :
where the iterated Pythagorean means have been identified with their parts in progressing order:
  • H is the harmonic mean,
  • HG is the harmonic–geometric mean,
  • G = HA is the geometric mean,
  • GA is the geometric–arithmetic mean,
  • A is the arithmetic mean.