Functional means

Functional means are a way of inducing a mean M_f from a differentiable function f.

$M_f(x, y) = (f')^{-1} \left( \frac{f(y) - f(x)}{y-x} \right)$

If f(x) = x², M_f(x, y) = (x + y)/2 is the arithmetic mean.

If f(x) = 1/x, M_f(x, y) = √(xy) is the geometric mean.

Seems like this should be useful for something.

Related post: Means and inequalities