TITLE: Prove that the $p$-mean is an increasing function of $p$
QUESTION [8 upvotes]: Let $p\neq0$ and $j=1,2,\cdots,n$ and $x_j>0$ and $$\chi(p)=\left(\frac{1}{n}\sum_{j=1}^nx_j^p\right)^\frac{1}{p}.$$ Prove that $\chi$ is strictly increasing and the following statements hold

$\lim\limits_{p\to0}\chi(p)=(x_1x_2\cdots x_n)^\frac{1}{n}$
$\lim\limits_{p\to+\infty}\chi(p)=\max\{x_1,x_2,\cdots, x_n\}$
$\lim\limits_{p\to-\infty}\chi(p)=\min\{x_1,x_2,\cdots, x_n\}$

I don't have any idea to prove them!

REPLY [2 votes]: The proofs you're looking for are in Chapter III (THE POWER MEANS) of Handbook of Means and Their Inequalities by P. S. Bullen [Kluwer, 2003].