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Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + \left|\left|f-g\right|\right|_p^p \leq 2^{p-1}\left(||f||_p^p + ||g||_p^p\right) $$ I have a couple of ideas, but I know they are wrong. I tried saying that $|f+g|^p \leq |f|^p + |g|^p$ and the same thing for $|f-g|$ but I know those inequalities aren't true in general.

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It is enough to prove that for each numbers $a$ and $b$, and $p\geqslant 2$, $$\left|\frac{a+b}2\right|^p+\left|\frac{a-b}2\right|^p\leqslant \frac12(|a|^p+|b|^p),$$ what was done here.

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