# Definition:Hölder Mean

## Definition

Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be positive real numbers.

Let $p$ be an extended real number.

The Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$ is denoted $\map {M_p} {x_1, x_2, \ldots, x_n}$.

### Non-Zero Exponent

For $p \in \R_{\ne 0}$, the Hölder mean is defined as:

$\ds \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}$

whenever the above expression is defined.

### Negative Exponent with Zero Parameter

For $p < 0$ and at least one $a_k = 0$, the Hölder mean is defined as:

$\ds \map {M_p} {x_1, x_2, \ldots, x_n} = 0$

### Zero Exponent

For $p = 0$, the Hölder mean is defined as:

$\map {M_0} {x_1, x_2, \ldots, x_n} = \paren {x_1 x_2 \cdots x_n}^{1 / n}$

which is the geometric mean of $x_1, x_2, \ldots, x_n$.

### Positive Infinite Exponent

For $p = \infty$, the Hölder mean is defined as:

$\map {M_\infty} {x_1, x_2, \ldots, x_n} = \max \set {x_1, x_2, \ldots, x_n}$

### Negative Infinite Exponent

For $p = -\infty$, the Hölder mean is defined as:

$\map {M_{-\infty} } {x_1, x_2, \ldots, x_n} = \min \set {x_1, x_2, \ldots, x_n}$

## Also known as

A Hölder mean is also known as a power mean or generalized mean.

Some sources denote it:

$\ds \map {\map M t} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^t}^{1 / t}$

## Also see

• Results about the Hölder mean can be found here.

## Source of Name

This entry was named for Otto Ludwig Hölder.