Definition:Hölder 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 $M_p \left({x_1, x_2, \ldots, x_n}\right)$.

For real $p \ne 0$, it is defined as:
 * $\displaystyle M_p \left({x_1, x_2, \ldots, x_n}\right) = \left({\frac 1 n \sum_{k \mathop = 1}^n x_k^p}\right)^{1/p}$

whenever the above expression is defined.

For $p = 0$, it is defined as:
 * $M_0 \left({x_1, x_2, \ldots, x_n}\right) = \left({x_1 x_2 \cdots x_n}\right)^{1/n}$

the geometric mean of $x_1, x_2, \ldots, x_n$.

For $p = \infty$, it is defined as:
 * $M_{\infty} \left({x_1, x_2, \ldots, x_n}\right) = \max {\left\{{x_1, x_2, \ldots, x_n}\right\}}$

For $p = -\infty$, it is defined as:
 * $M_{-\infty} \left({x_1, x_2, \ldots, x_n}\right) = \min {\left\{{x_1, x_2, \ldots, x_n}\right\}}$

Also known as
A Hölder mean is also known as a power mean or generalized mean.

Some sources denote it:
 * $\displaystyle M \left({t}\right) \left({x_1, x_2, \ldots, x_n}\right) = \left({\frac 1 n \sum_{k \mathop = 1}^n x_k^t}\right)^{1/t}$

Also see

 * Inequality of Hölder Means


 * Definition:Arithmetic Mean
 * Definition:Geometric Mean
 * Definition:Harmonic Mean


 * $p$-Norm