Definition:Hölder Mean

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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.

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.


Sources

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