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.
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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Generalized Mean: $3.1.14$