Limit of Hölder Mean as Exponent tends to Zero is Geometric 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:

$\ds \lim_{p \mathop \to 0} \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {x_1 x_2 \cdots x_n}^{1 / n}$

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


Proof

Let $p \in \R$ such that $p \ne 0$.

\(\ds \map {M_p} {x_1, x_2, \ldots, x_n}\) \(=\) \(\ds \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}\) Definition of Hölder Mean
\(\ds \leadsto \ \ \) \(\ds \map \ln {\map {M_p} {x_1, x_2, \ldots, x_n} }\) \(=\) \(\ds \map \ln {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}\) taking logarithm of both sides
\(\ds \) \(=\) \(\ds \dfrac {\map \ln {\dfrac 1 n \ds \sum_{k \mathop = 1}^n {x_k}^p} } p\) Logarithm of Power
\(\ds \leadsto \ \ \) \(\ds \map {M_p} {x_1, x_2, \ldots, x_n}\) \(=\) \(\ds \map \exp {\dfrac {\map \ln {\dfrac 1 n \ds \sum_{k \mathop = 1}^n {x_k}^p} } p}\) taking exponential of both sides


With a view to using L'Hôpital's Rule, let us express the argument of the exponential on the right hand side in the form:

\(\ds \dfrac {\map \ln {\dfrac 1 n \ds \sum_{k \mathop = 1}^n {x_k}^p} } p\) \(=\) \(\ds \dfrac {\map f p} {\map g p}\) where $\map f p := \map \ln {\dfrac 1 n \ds \sum_{k \mathop = 1}^n {x_k}^p}$ and $\map g p := p$


Then we have:

\(\ds \map {f'} p\) \(=\) \(\ds \map {\dfrac \d {\d p} } {\map \ln {\dfrac 1 n \sum_{k \mathop = 1}^n {x_k}^p} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 {\dfrac 1 n \ds \sum_{k \mathop = 1}^n {x_k}^p} \paren {\dfrac 1 n \ds \sum_{k \mathop = 1}^n {x_k}^p \ln x_k}\) Derivative of Natural Logarithm Function, Derivative of General Exponential Function, Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac {\ds \sum_{k \mathop = 1}^n {x_k}^p \ln x_k} {\ds \sum_{k \mathop = 1}^n {x_k}^p}\) simplifying

and:

\(\ds \map {g'} p\) \(=\) \(\ds \map {\dfrac \d {\d p} } p\)
\(\ds \) \(=\) \(\ds 1\) Derivative of Identity Function


Hence:

\(\ds \lim_{p \mathop \to 0} \dfrac {\map f p} {\map g p}\) \(=\) \(\ds \lim_{p \mathop \to 0} \dfrac {\map {f'} p} {\map {g'} p}\) L'Hôpital's Rule
\(\ds \) \(=\) \(\ds \lim_{p \mathop \to 0} \dfrac {\paren {\dfrac {\ds \sum_{k \mathop = 1}^n {x_k}^p \ln x_k} {\ds \sum_{k \mathop = 1}^n {x_k}^p} } } 1\) substituting for $f'$ and $g'$
\(\ds \) \(=\) \(\ds \lim_{p \mathop \to 0} \dfrac {\ds \sum_{k \mathop = 1}^n {x_k}^p \ln x_k} {\ds \sum_{k \mathop = 1}^n {x_k}^p}\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {\ds \sum_{k \mathop = 1}^n {x_k}^0 \ln x_k} {\ds \sum_{k \mathop = 1}^n {x_k}^0}\) letting $p \to 0$
\(\ds \) \(=\) \(\ds \dfrac {\ds \sum_{k \mathop = 1}^n 1 \ln x_k} {\ds \sum_{k \mathop = 1}^n 1}\) Zeroth Power of Real Number equals One
\(\ds \) \(=\) \(\ds \dfrac 1 n \sum_{k \mathop = 1} \ln x_k\) further simplification
\(\ds \) \(=\) \(\ds \dfrac 1 n \ln \prod_{k \mathop = 1} x_k\) Sum of Logarithms
\(\ds \) \(=\) \(\ds \map \ln {\paren {\prod_{k \mathop = 1} x_k}^{1 / n} }\) Logarithm of Power
\(\ds \leadsto \ \ \) \(\ds \lim_{p \mathop \to 0} \map {M_p} {x_1, x_2, \ldots, x_n}\) \(=\) \(\ds \map \exp {\map \ln {\paren {\prod_{k \mathop = 1} x_k}^{1 / n} } }\)
\(\ds \) \(=\) \(\ds \paren {\prod_{k \mathop = 1} x_k}^{1 / n}\)
\(\ds \) \(=\) \(\ds \paren {x_1 x_2 \cdots x_n}^{1 / n}\)

$\blacksquare$


Sources

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