User:InfoTheorist/Power means inequality

In this page we give a proof of the power means inequality for powers which are positive integers using induction (the same inequality follows for negative integers by applying the inequality to reciprocals of the original numbers). Let $x_{1},\dots ,x_{n}$ be positive real numbers, $w_{1},\dots ,w_{n}$ be non-negative weights that add up to one, and let k be a positive integer. First note that by the Cauchy-Schwarz inequality,

${\frac {\sum _{i=1}^{n}w_{i}x_{i}^{k}}{\sum _{i=1}^{n}w_{i}x_{i}^{k-1}}}\leq {\frac {\sum _{i=1}^{n}w_{i}x_{i}^{k+1}}{\sum _{i=1}^{n}w_{i}x_{i}^{k}}},$

with equality if and only if all the $x_{i}$'s are equal. The expression on the left hand side is known as the weighted Lehmer mean and is denoted by $L_{k,w}(x)$.

Our aim is to show

$\left(\sum _{i=1}^{n}w_{i}x_{i}^{k}\right)^{\frac {1}{k}}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{k+1}\right)^{\frac {1}{k+1}}.$

Note that for k=1 this inequality follows easily from Cauchy-Schwarz since

$\left(\sum _{i=1}^{n}w_{i}x_{i}\right)^{2}\leq \left(\sum _{i=1}^{n}w_{i}\right)\left(\sum _{i=1}^{n}w_{i}x_{i}^{2}\right).$

Now suppose the inequality is true for a particular k and we show that it's true for k+1. Thus

$\left(\sum _{i=1}^{n}w_{i}x_{i}^{k}\right)^{\frac {1}{k}}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{k+1}\right)^{\frac {1}{k+1}}.$

Raising both sides to the k(k+1) power and simplifying results in

$\sum _{i=1}^{n}w_{i}x_{i}^{k}\leq \left({\frac {\sum _{i=1}^{n}w_{i}x_{i}^{k+1}}{\sum _{i=1}^{n}w_{i}x_{i}^{k}}}\right)^{k}.$

Multiplying both sides by $L_{k,w}(x)$ we get

$\sum _{i=1}^{n}w_{i}x_{i}^{k+1}\leq \left({\frac {\sum _{i=1}^{n}w_{i}x_{i}^{k+1}}{\sum _{i=1}^{n}w_{i}x_{i}^{k}}}\right)^{k+1}.$

Now using the fact that

$L_{k,w}(x)\leq L_{k+1,w}(x)$

results in

$\sum _{i=1}^{n}w_{i}x_{i}^{k+1}\leq \left({\frac {\sum _{i=1}^{n}w_{i}x_{i}^{k+2}}{\sum _{i=1}^{n}w_{i}x_{i}^{k+1}}}\right)^{k+1},$

which can simplified to the desired inequality

$\left(\sum _{i=1}^{n}w_{i}x_{i}^{k+1}\right)^{\frac {1}{k+1}}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{k+2}\right)^{\frac {1}{k+2}}.$