Let $f:[\frac{\delta}{2},\delta] \to \mathbb{R}^{+}$, continuous. Here, $\delta >0$ small. Define, $F(x) = \int_{\frac{\delta}{2}}^{x}f(s)ds$. Note that $F^{\prime}(x) = f(x)$. What conditions do I have to put on the function $f$ to show that there exists a constant $C > 0$ such that $$ \int_{\frac{\delta}{2}}^{\delta}\dfrac{F^{\prime}(x)}{F(\delta)}|g(x)|^{2}dx \leq C ? $$ where $g \in L^{2}\big(0,1)$.
If $f$ is increasing, does it work? That's all, I'm not getting it.