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

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

$\blacksquare$