# Generalized mean

If *t* is a non-zero real number, we can define the **generalized mean with exponent ***t* of the positive real numbers *a*_{1},...,*a*_{n} as

The case

*t* = 1 yields the

arithmetic mean and the case

*t* = -1 yields the

harmonic mean. As

*t* approaches 0, the

limit of M(

*t*) is the

geometric mean of the given numbers, and so it makes sense to

*define* M(0) to be the geometric mean. Furthermore, as

*t* approaches ∞, M(

*t*) approaches the maximum of the given numbers, and as

*t* approaches -∞, M(

*t*) approaches the minimum of the given numbers.

In general, if -∞ <= *s* < *t* <= ∞, then

- M(
*s*) <= M(*t*)

and the two means are equal if and only if

*a*_{1} =

*a*_{2} = ... =

*a*_{n}. Furthermore, if

*a* is a positive real number, then the generalized mean with exponent

*t* of the numbers

*aa*_{1},...,

*aa*_{n} is equal to

*a* times the generalized mean of the numbers

*a*_{1},...,

*a'\'*_{}n''.

This could be generalized further as

and again a suitable choice of an invertible f(

*x*) will give the arithmetic mean with f(

*x*)=

*x*, the geometric mean with f(

*x*)=log(

*x*), the harmonic mean with f(

*x*)=1/

*x*, and the generalized mean with exponent

*t* with f(

*x*)=

*x*^{t}. But other functions could be used, such as f(

*x*)=e

^{x}.

**See also:** average