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How do I prove this using Jensen's inequality?

$$\biggl|\prod_{i=1}^n x_{i}\biggl|^p \le {n^{p-1}}\sum_{i=1}^n |x_{i}|^p$$

I've tried log on both sides but I couldn't find a common expression.

on the right side I did: $$log( E(n|x|)^p) = log({n^{p-1}}\sum_{i=1}^n |x_{i}|^p)$$ and on the left side: $$ E(log(|x|)^{np}) = log\bigg(\bigg|\prod_{i=1}^n x_{i}\bigg|\bigg)^p$$

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1 Answer 1

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It's wrong.

Try $p=1$, $n=2$ and $a=b=3$.

We need to prove that $6\geq9$, which is not so true.

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