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 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:

:<math>g_{n+1} = \sqrt{g_n h_n}</math>

and

:<math>h_{n+1} = \frac{2{g_n}{h_n{g_n + h_n}</math>

Both of these sequences converge to the same number, which we call the geometric–harmonic mean M(x,&nbsp;y) of x and&nbsp;y. The geometric–harmonic mean is also designated as the harmonic–geometric mean.&nbsp;(cf. Wolfram MathWorld below.)

The existence of the limit can be proved by the means of Bolzano&ndash;Weierstrass theorem in a manner almost identical to the proof of existence of arithmetic–geometric mean.

Properties

M(x,&nbsp;y) is a number between the geometric and harmonic mean of x and y; in particular it is between x and y. M(x,&nbsp;y) is also homogeneous, i.e. if r&nbsp;>&nbsp;0, then M(rx,&nbsp;ry) =&nbsp;r M(x,&nbsp;y).

If AG(x, y) is the arithmetic–geometric mean, then we also have

:<math>M(x,y) = \frac{1}{AG(\frac{1}{x},\frac{1}{y})}</math>

Inequalities

We have the following inequality involving the Pythagorean means {H,&nbsp;G,&nbsp;A} and iterated Pythagorean means {HG,&nbsp;HA,&nbsp;GA}:

:<math>\min(x,y) \leq H(x,y) \leq HG(x,y) \leq G(x,y) \leq GA(x,y) \leq A(x,y) \leq \max(x,y)</math>

where the iterated Pythagorean means have been identified with their parts {H,&nbsp;G,&nbsp;A} in progressing order:

  • H(x,&nbsp;y) is the harmonic mean,
  • HG(x,&nbsp;y) is the harmonic–geometric mean,
  • G(x,&nbsp;y) =&nbsp;HA(x,&nbsp;y) is the geometric mean (which is also the harmonic–arithmetic mean),
  • GA(x,&nbsp;y) is the geometric–arithmetic mean,
  • A(x,&nbsp;y) is the arithmetic mean.

See also

  • Arithmetic–geometric mean
  • Arithmetic–harmonic mean
  • Mean