Prove that $\|f+g\|_4^4+\|f-g\|_4^4 \leq (\|f\|_4+\|g\|_4)^4 + (\|f\|_4-\|g\|_4)^4$ for $f,g :\mathbb{R}^d \rightarrow \mathbb{R}$.
Notice that you can't solve it with minkowski's inequality.
My approach:
Right part: By expanding the two terms I get $2\|f\|_4^4+12\|f\|_4^2\|g\|_4^2+2\|g\|_4^4$.
Left part: $\|f+g\|_4^4+\|f-g\|_4^4 = \int|f+g|^4 + \int|f-g|^4=\int |f+g|^4 +|f-g|^4 = 2\|f\|_4^4+12\int|f|^2|g|^2+2\|g\|_4^4 $.
So I have to compare $\int|f|^2|g|^2$ with $\|f\|_4^2\|g\|_4^2$ and by Holder's inequality for $p=2$, $q=2$. I am done.
Is my approach correct? Is there a faster solution? Thank you in advance!!!!
You approach looks correct to me. Notice that an application of the inequality in this question with $p=2$ does not give the result you want. Therefore, I am not sure there is a faster solution.