Geometric-harmonic mean

In mathematics, the geometric-harmonic mean M("x", "y") of two positive real numbers "x" and "y" is defined as follows: we first 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 then 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".

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 sequences ("g"_"n") and ("h"_"n") are defined:

:$g\_\{n+1\}\; =\; sqrt\{g\_n\; h\_n\}$

and

:$h\_\{n+1\}\; =\; frac\{2\}\{frac\{1\}\{g\_n\}\; +\; frac\{1\}\{h\_n$

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

:$M(x,y)\; =\; frac\{1\}\{AG(frac\{1\}\{x\},frac\{1\}\{y\})\}$

ee also

*arithmetic-geometric mean
*arithmetic-harmonic mean
*mean

External links

* [http://mathworld.wolfram.com/Harmonic-GeometricMean.html Harmonic-Geometric Mean on Mathworld]