$f, f^2 \in L^2[0,1]$ implies $f+1 \in L^3[0,1]$

Question

Suppose $f,f^2 \in L^2[0,1]$ and show $f+1 \in L^3[0,1]$.

So I know if $f,f^2 \in L^2[0,1]$, then

$$\bigg(\int_{[0,1]} \vert f \vert^2 dx\bigg)^\frac{1}{2}<+\infty$$

and

$$\bigg(\int_{[0,1]} \vert f^2 \vert^2 dx\bigg)^\frac{1}{2}<+\infty$$

So how do I bound $\bigg(\int_{[0,1]} \vert f+1 \vert^3 dx \bigg)^\frac{1}{3}$

to be less than $\infty$?

Answer 1

By Minkowski, it suffices to show $f \in L^3$. By Hölder, $\|f^3\|_{1} \leq \|f^2\|_{2}\|f\|_{2}<\infty$ by assumption