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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

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

I don't have any idea to prove them!

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    $\begingroup$ en.wikipedia.org/wiki/Generalized_mean - there you go. these are called $p$-means, you can just apply what wikipedia says (it provides all necessary proofs) with equal weights $1/n$ $\endgroup$ – mm-aops Jun 13 '14 at 16:37
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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].

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