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 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 (in any order) 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 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. 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

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

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

Inequalities

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

$\min(x,y) \leq H(x,y) \leq HG(x,y) \leq G(x,y) \leq GA(x,y) \leq A(x,y) \leq \max(x,y)$

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.

See also

• Arithmetic-geometric mean
• Arithmetic-harmonic mean
• Mean

External links

• Weisstein, Eric W., "Harmonic-Geometric Mean" from MathWorld.