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 $\map {M_p} {x_1, x_2, \ldots, x_n}$.
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:
- $\map {M_{-\infty} } {x_1, x_2, \ldots, x_n} = \min {\set {x_1, x_2, \ldots, x_n} }$
Also known as
A Hölder mean is also known as a power mean or generalized mean.
Some sources denote it:
- $\displaystyle \map {\map M t} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n x_k^t}^{1/t}$
Also see
Source of Name
This entry was named for Otto Ludwig Hölder.
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3.1.14$: Generalized Mean