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!

  • 2
    $\begingroup$ 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

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].


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.