Definition:Hölder Mean/Zero Exponent

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Definition

Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be positive real numbers.

Let $p$ be an extended real number.


Let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.


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 $x_1, x_2, \ldots, x_n$.


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