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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Generalized Mean: $3.1.18$