Let $f$ be a $L^{p}$ function on $\mathbb{R}$. If $p>4/3$, prove that:
$\lim_{t\rightarrow0^{+}}\int_{0}^{t}x^{-1/4}f(x)dx=0.$
The natural procedure here is to bound the integral by using Hölder's inequality; after using the Hölder inequality, we get that as $t\rightarrow0^{+}$ the integral from $0$ to $t$ of $\left|x^{-1/4}f(x)\right|$ goes to zero. Does the desired result follow immediately from the Dominated Convergence Theorem?