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

$\backslash bar\{x\}=f^\{-1\}\backslash left(\; \backslash frac\{f(x\_1)+f(x\_2)\}2\; \backslash right).$

For n numbers

x₁, ..., x_n in S,

the f-mean is

$\backslash bar\{x\}=f^\{-1\}\backslash left(\; \backslash frac\{f(x\_1)+\; \backslash cdots\; +\; f(x\_n)\}n\; \backslash right).$

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

$\backslash frac\{f\backslash left(x\_1\backslash right)\; +\; f\backslash left(x\_2\backslash right)\}2$

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

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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 as long as it is positive and not 1.

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

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• Generalized mean
• Jensen's inequality

Means