Generalised f-mean
From Academic Kids
In mathematics and statistics, the generalised f-mean is the natural generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f(x).
If f is a function which maps a connected subset S of the real line to the real numbers, and is both continuous and injective then we can define the f-mean of two numbers
- x1, x2 in S
as
- <math>\overline{x}=f^{-1}( (f(x_1)+f(x_2))/2 ).<math>
For n numbers
- x1, ..., xn in S,
the f-mean is
- <math>\overline{x}=f^{-1}( (f(x_n)+ \cdots + f(x_n))/n ).<math>
We require f to be injective in order for the inverse function f −1 to exist. Continuity is required to ensure
- <math>\left(f\left(x_1\right) + f\left(x_2\right)\right)/2<math>
lies within the domain of f -1.
Since f is injective and continuous, it follows that f is a strictly monotonic function, and therefore that the f-mean is neither larger than the largest number in {xi} nor smaller than the smallest number in {xi}.
Examples
If we take S to be the real line and f(x) = x, then the f-mean corresponds to the arithmetic mean.
If we take S to be the set of positive real numbers and f(x) = log(x), then the f-mean corresponds to the geometric mean. The result does not depend on the base of the logarithm.
If we take S to be the set of positive real numbers and f(x) = 1/x, then the f-mean corresponds to the harmonic mean.
