Let $f:[0,1]\rightarrow \mathbb{R}$ be an absolute continuous function, such as $f' \in L^4(0,1)$ and $f(0)=0$. Find $r<0$, such as, for all $\alpha \geq r,$ $lim_{x\rightarrow 0^{+}} x^{\alpha}f(x)=0.$
Things I know: 1) $f$ is absolute continuous, so $f'$ exists, $f'\in L^1(0,1)$ and $f(x)=f(0)+\int_{0}^{x} f'(x) dx=\int_{0}^{x} f'(x) dx$ for all $x \in [0,1]$.
2) Using Jensen inequality we have $|f(x)|^4\leq (\int_{0}^{x} |f'(x)| dx)^4 \leq \int_{0}^{x} |f'(x)|^4 dx\leq \int_{0}^{1} |f'(x)|^4 dx$. So $|f(x)|\leq ||f'||_{4}$ for all $x \in [0,1]$
3) $x^{\alpha}|f(x)|\leq x^{\alpha} ||f'||_{4}$
Now I'm stuck, I think I should use that $\int_{0}^{1} 1/x^{\alpha} dx<\infty$ if $\alpha>1$ but I don't know what to do. Thanks a lot for your help!!