generalization of Young inequality

It’s straightforward to extend Young inequality (http://planetmath.org/YoungInequality) to an arbitrary finite number of : provided that $a_{i}>0$ , $c_{i}>0$ and $\sum_{i=1}^{n}\frac{1}{c_{i}}=\frac{1}{r}$ ,

\left(\prod_{i=1}^{n}a_{i}\right)^{r}\leq r\sum_{i=1}^{n}\frac{a_{i}^{c_{i}}}{% c_{i}}

In fact,

$\displaystyle\left(\prod_{i=1}^{n}a_{i}\right)^{r}$	$\displaystyle=$	$\displaystyle\exp\left[\log\left(\prod_{i=1}^{n}a_{i}\right)^{r}\right]$
	$\displaystyle=$	$\displaystyle\exp\left[r\sum_{i=1}^{n}\log a_{i}\right]$
	$\displaystyle=$	$\displaystyle\exp\left[r\sum_{i=1}^{n}\frac{1}{c_{i}}\log\left(a_{i}^{c_{i}}% \right)\right]$
	$\displaystyle=$	$\displaystyle\exp\left[\frac{\sum_{i=1}^{n}\frac{1}{c_{i}}\log\left(a_{i}^{c_{% i}}\right)}{\frac{1}{r}}\right]$
(by Jensen’s inequality and monotonicity of exp)	$\displaystyle\leq$	$\displaystyle\exp\left[\log\left(\frac{\sum_{i=1}^{n}\frac{1}{c_{i}}a_{i}^{c_{% i}}}{\frac{1}{r}}\right)\right]$
		$\displaystyle=r\sum_{i=1}^{n}\frac{a_{i}^{c_{i}}}{c_{i}}$

Remark: in the case

\frac{1}{c_{i}}=1\text{ \ \ \ \ }\forall i

one obtains:

\left(\prod_{i=1}^{n}a_{i}\right)^{\frac{1}{n}}\leq\frac{1}{n}\sum_{i=1}^{n}a_% {i}

that is, the usual arithmetic-geometric mean inequality, which suggests Young inequality could be regarded as a generalization of this classical result. Actually, let’s consider the following restatement of Young inequality. Having defined: $w_{i}=\frac{1}{c_{i}}$ , $\ \ \sum_{i=1}^{n}w_{i}=W=\frac{1}{r}$ , $\ \ \ x_{i}=a_{i}^{\frac{1}{w_{i}}}$ we have:

\left(\prod_{i=1}^{n}x_{i}^{w_{i}}\right)^{\frac{1}{W}}\leq\frac{1}{W}\sum_{i=% 1}^{n}w_{i}x_{i}

This expression shows that Young inequality is nothing else than geometric-arithmentic weighted mean (http://planetmath.org/ArithmeticMean) inequality.

Title	generalization of Young inequality
Canonical name	GeneralizationOfYoungInequality
Date of creation	2013-03-22 15:43:08
Last modified on	2013-03-22 15:43:08
Owner	Andrea Ambrosio (7332)
Last modified by	Andrea Ambrosio (7332)
Numerical id	25
Author	Andrea Ambrosio (7332)
Entry type	Result
Classification	msc 46E30