derivation of zeroth weighted power mean

Let $x_{1},x_{2},\ldots,x_{n}$ be positive real numbers, and let $w_{1},w_{2},\ldots,w_{n}$ be positive real numbers such that $w_{1}+w_{2}+\cdots+w_{n}=1$ . For $r\neq 0$ , the $r$ -th weighted power mean of $x_{1},x_{2},\ldots,x_{n}$ is

$M_{w}^{r}(x_{1},x_{2},\ldots,x_{n})=(w_{1}x_{1}^{r}+w_{2}x_{2}^{r}+\cdots+w_{n% }x_{n}^{r})^{1/r}.$

Using the Taylor series expansion $e^{t}=1+t+{\mathcal{O}}(t^{2})$ , where ${\mathcal{O}}(t^{2})$ is Landau notation for terms of order $t^{2}$ and higher, we can write $x_{i}^{r}$ as

$x_{i}^{r}=e^{r\log x_{i}}=1+r\log x_{i}+{\mathcal{O}}(r^{2}).$

By substituting this into the definition of $M_{w}^{r}$ , we get

$\displaystyle M_{w}^{r}(x_{1},x_{2},\ldots,x_{n})$	$\displaystyle=$	$\displaystyle\left[w_{1}(1+r\log x_{1})+\cdots+w_{n}(1+r\log x_{n})+{\mathcal{% O}}(r^{2})\right]^{1/r}$
	$\displaystyle=$	$\displaystyle\left[1+r(w_{1}\log x_{1}+\cdots+w_{n}\log x_{n})+{\mathcal{O}}(r% ^{2})\right]^{1/r}$
	$\displaystyle=$	$\displaystyle\left[1+r\log(x_{1}^{w_{1}}x_{2}^{w_{2}}\cdots x_{n}^{w_{n}})+{% \mathcal{O}}(r^{2})\right]^{1/r}$
	$\displaystyle=$	$\displaystyle\exp\left\{\frac{1}{r}\log\left[1+r\log(x_{1}^{w_{1}}x_{2}^{w_{2}% }\cdots x_{n}^{w_{n}})+{\mathcal{O}}(r^{2})\right]\right\}.$

Again using a Taylor series, this time $\log(1+t)=t+{\mathcal{O}}(t^{2})$ , we get

	$\displaystyle M_{w}^{r}(x_{1},x_{2},\ldots,x_{n})$	$\displaystyle=$	$\displaystyle\exp\left\{\frac{1}{r}\left[r\log(x_{1}^{w_{1}}x_{2}^{w_{2}}% \cdots x_{n}^{w_{n}})+{\mathcal{O}}(r^{2})\right]\right\}$
		$\displaystyle=$	$\displaystyle\exp\left[\log(x_{1}^{w_{1}}x_{2}^{w_{2}}\cdots x_{n}^{w_{n}})+{% \mathcal{O}}(r)\right].$

Taking the limit $r\to 0$ , we find

	$\displaystyle M_{w}^{0}(x_{1},x_{2},\ldots,x_{n})$	$\displaystyle=$	$\displaystyle\exp\left[\log(x_{1}^{w_{1}}x_{2}^{w_{2}}\cdots x_{n}^{w_{n}})\right]$
		$\displaystyle=$	$\displaystyle x_{1}^{w_{1}}x_{2}^{w_{2}}\cdots x_{n}^{w_{n}}.$

In particular, if we choose all the weights to be $\frac{1}{n}$ ,

$M^{0}(x_{1},x_{2},\ldots,x_{n})=\sqrt[n]{x_{1}x_{2}\cdots x_{n}},$

the geometric mean of $x_{1},x_{2},\ldots,x_{n}$ .

Title	derivation of zeroth weighted power mean
Canonical name	DerivationOfZerothWeightedPowerMean
Date of creation	2013-03-22 13:10:29
Last modified on	2013-03-22 13:10:29
Owner	pbruin (1001)
Last modified by	pbruin (1001)
Numerical id	6
Author	pbruin (1001)
Entry type	Derivation
Classification	msc 26B99
Related topic	PowerMean
Related topic	GeometricMean
Related topic	GeneralMeansInequality
Related topic	DerivationOfHarmonicMeanAsTheLimitOfThePowerMean