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Generalised f-mean

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

$\overline{x}=f^{-1}( (f(x_1)+f(x_2))/2 ).$

For n numbers

x₁, ..., x_n in S,

the f-mean is

$\overline{x}=f^{-1}( (f(x_n)+ \cdots + f(x_n))/n ).$

We require f to be injective in order for the inverse function f^-1 to exist. Continuity is required to ensure

$\left(f\left(x_1\right) + f\left(x_2\right)\right)/2$

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 {x_i} nor smaller than the smallest number in {x_i}.

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.

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