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Let $f$ be continuous, nonzero on $[a,b]$. For $p\in \mathbb R$, define $$\Phi_p(f)=\left(\frac{1}{b-a}\int_a^b|f(x)|^pdx\right)^{1/p}$$.

Show that $$\lim_{p\to 0} \Phi_p(f)=\exp(\frac{1}{b-a}\int_a^b \ln |f(x)|dx)$$

I have no idea on it. Once I take L' Hospital formula, I got nonsense....

xldd
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  • See this answer in a more general setting : http://math.stackexchange.com/questions/282271/scaled-lp-norm-and-geometric-mean/282311#282311 and take a look to the related questions. – user37238 Jul 02 '14 at 09:42
  • For $\log |x|$, $|x|^p$, $0<p<1$ are concave functions. You should be able to prove this using Jensen's inequality. – J.R. Jul 02 '14 at 09:44

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