I have two problems here.
1) I need to show $\|xf(x)\|_2 \le \|f(x)\|_3$ given $f \in L^3[0,1]$.
My approach: I know $\|f(x)\|_2 \le \|f(x)\|_3$, and hope to show $\|xf(x)\|_2 \le \|f(x)\|_2$. That's true because $x \in [0,1]$, $xf(x) \le f(x)$. Am I right?
2) If $f \in L^{5}(E)$, $E=[0,1]$ and $\int_{E} f(x)dx=0$, then $\int_{E} |1-f|^5 dx \ge 1$.
I think the key is to define $g=1-f$ and use $\|g\|_p \ge \|g\|_2$ for $p \ge 2$. Please give me any confirmation or tell me wrong.