TITLE: Two $L^p$ space problems
QUESTION [6 upvotes]: 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.

REPLY [4 votes]: For (1), we can apply Hölder's inequality:
$$\|xf\|_{L^2[0,1]}=\sqrt{\int_{[0,1]}x^2f^2}\leq\sqrt{\Big(\int_{[0,1]}x^{2\cdot3}\Big)^{\frac{1}{3}}\Big(\int_{[0,1]}f^{2\cdot\frac{3}{2}}\Big)^{\frac{2}{3}}}=\Big(\int_{[0,1]}x^{6}\Big)^{\frac{1}{6}}\|f\|_{L^3[0,1]}.$$
Now (1) follow from 
$$\int_{[0,1]}x^{6}=\frac{x^7}{7}\Big|_0^1=\frac{1}{7}\leq 1.$$
For (2), since $\displaystyle\int_{[0,1]}f=0$ by assumption, we have
$$1=\int_{[0,1]}1=\int_{[0,1]}(1-f).$$
Now apply the Hölder's inequality to the right hand side, we obtain
$$1=\int_{[0,1]}(1-f)\leq \Big(\int_{[0,1]}|1-f|^5\Big)^{\frac{1}{5}}\Big(\int_{[0,1]}1\Big)^{\frac{4}{5}}=\Big(\int_{[0,1]}|1-f|^5\Big)^{\frac{1}{5}},$$
which implies that
$$1\leq \int_{[0,1]}|1-f|^5,$$
as required.