Definition:Hölder Mean

This page is about Hölder Mean. For other uses, see Mean.

Definition

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

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

For real $p \ne 0$, it is defined as:

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

whenever the above expression is defined.

For $p = 0$, it is defined as:

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

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

For $p = \infty$, it is defined as:

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

For $p = -\infty$, it is defined as:

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

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

Some sources denote it:

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

This entry was named for Otto Ludwig Hölder.