Hölder Mean for Exponent 1 is Arithmetic Mean

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Theorem

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

For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.


Then:

$\map {M_1} {x_1, x_2, \ldots, x_n} = \dfrac {x_1 + x_2 + \cdots + x_n} n$

which is the arithmetic mean of $x_1, x_2, \ldots, x_n$.


Proof

Recall the definition of the Hölder mean with exponent $p$:

$\ds \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}$

Then:

\(\ds \map {M_1} {x_1, x_2, \ldots, x_n}\) \(=\) \(\ds \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^1}^{1 / 1}\)
\(\ds \) \(=\) \(\ds \frac 1 n \sum_{k \mathop = 1}^n {x_k}\) simplifying

which is the arithmetic mean by definition.

$\blacksquare$


Sources

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