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Generalised f-mean

In mathematics and statistics, the generalised f-mean is the natural generalisation of the more familar 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, ..., x_n 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

(f(x₁) + f(x₂))/2

lies within the domain of f ^-1.

Since f is injective and continuous, it follows that f is strictly increasing; 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}.

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

(the natural logarithm), then the f-mean corresponds to the geometric mean.

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.

See also: Jensen's inequality.