Sorry if this is a silly question but I want to confirm a result. I get some kind of inverted Minkowski's inequality for $p\in[2,\infty )$. It is based in Clarkson's inequality what can be written as
$$ 2(\|f\|_p^p+\|g\|_p^p)\leqslant \|f+g\|_p^p+\|f-g\|_p^p\tag1 $$ for $f,g\in L^p(\mu )$. But then we reach the conclusion that $$ \|f\|_p^p+\|g\|_p^p\leqslant \||f|+|g|\|_p^p\tag2 $$ This inequality seems too good to be correct. Can someone check if there is no weird mistake in this result? Thank you.
No, this is perfectly fine (and rather easy to prove). For $p\ge 1$, the map $x\mapsto x^p$ is convex on $[0,\infty)$. Since it sends $0$ to $0$, it is super-additive on $[0,\infty)$: in fact, by convexity, for all $x\ge 0$ and $y>0$,
$$\frac{(x+y)^p-x^p}{y}\ge \frac{y^p-0^p}y$$
Therefore $$\int(\lvert f\rvert+\lvert g\rvert)^p\ge\int (\lvert f\rvert^p+\lvert g\rvert^p)$$
In fact, when you go on to prove Minkowski, main part of the work is getting rid of those absolute values that just give a weaker reverse inequality.