# Geometric–harmonic mean

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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*_{0} = *x* and *h*_{0} = *y* and call it *g*_{1}, i.e. *g*_{1} is the square root of *xy*. We also form the harmonic mean of *x* and *y* and call it *h*_{1}, i.e. *h*_{1} 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*_{1} taking the place of *x* and *h*_{1} taking the place of *y*. In this way, two interdependent sequences (*g*_{n}) and (*h*_{n}) are defined:

and

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.

## Properties[edit]

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

## Inequalities[edit]

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

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

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