\begin{definition}[Definition: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 $\map {M_p} {x_1, x_2, \ldots, x_n}$.
For real $p \ne 0$, it 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.
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:
:$\map {M_{-\infty} } {x_1, x_2, \ldots, x_n} = \min \set {x_1, x_2, \ldots, x_n}$
\end{definition}