Category:Hölder Mean
This category contains results about the Hölder mean.
Definitions specific to this category can be found in Definitions/Hölder Mean.
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:
- $\map {M_{-\infty} } {x_1, x_2, \ldots, x_n} = \min \set {x_1, x_2, \ldots, x_n}$
Pages in category "Hölder Mean"
The following 8 pages are in this category, out of 8 total.