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$f_1,\ldots,f_n$ are positive functions. Set $F_j(x)=\int_0^xf_j(w)dw$ for each $j.$

How can one prove that

$$ \int_0^{+\infty}{\left(\frac{F_1(x)F_2(x)\ldots F_n(x)}{x^n}\right)}^{\frac{p}{n}}dx\leq \left(\frac{p}{p-1}\right)^p\int_0^{+\infty} (f_1(x)+f_2(x)+\ldots+f_n(x))^p dx ?$$

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What do you know about $p$? – Davide Giraudo Jul 3 '11 at 10:05
up vote 1 down vote accepted

The Hardy's inequality plus the AM–GM inequality.

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