Let $E \subset \mathbb{R}$ be a measurable subset. Assume that $\int_{E} |x|^{1/4} |f(x)|^2 dx < \infty$ and $\int_{E} x^4 |f(x)|^3 dx < \infty$, then I want to prove $f \in L^1 (E)$.
Morally the first inequality says that $f(x)$ behaves nicely at $0$ and the second inequality says that $f(x)$ behaves nicely at $\pm \infty$ but I'm struggling to prove the statement rigorously. A possible idea is to use Holder's inequality: we know that $f(x) x^{1/8} \in L^2 (E)$ and $f(x) x^{4/3} \in L^3 (E)$ and probably it can be used somehow but I don't know how. Anyways, any ideas are greatly appreciated!
1. On $\{x\in E: |x|\le 1\}$, write $|f(x)| = \left(|f(x)| \cdot|x|^{1/8}\right)\cdot |x|^{-1/8}$ and apply the Cauchy-Schwarz inequality.
2. On $\{x\in E: |x|>1\}$, write $|f(x)| = \left(|f(x)| \cdot |x|^{4/3}\right)\cdot |x|^{-4/3}$ and apply Hölder with the conjugate exponents $p=3$, $q=3/2$.