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

*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*)

*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

*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