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 x and y and call it g₁, i.e. g₁ is the square root of xy. We then 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. 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 (g_n) and (h_n) are defined:

g_{n+1} = \sqrt{a_n g_n}

and

h_{n+1} = \frac{2}{\frac{1}{a_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).

Geometric-harmonic Mean

See also