Let $\mathbb{D}\subset \mathbb{C}$ denote the unit disk.
I would like to seek sufficient—and, if possible, necessary—conditions on a weight function $$\mu: \mathbb{D} \rightarrow (0,+\infty),\quad \mu \in C^1(\overline{\mathbb{D}}),$$ for which there exists a constant $\delta >0$ with the following property:
For every analytic function $h_1$ on $\mathbb{D}$, there exists an analytic function $h_2$ on $\mathbb{D}$ such that $$2 \text{Re} \int_{\mathbb{D}} h_1 \overline{h}_2 \geq \delta \int_{\mathbb{D}} |h_1|^2 \mu + \int_{\mathbb{D}} |h_2|^2 \frac{1}{\mu}.$$
It can be seen that the property holds when $\mu$ is a constant. I therefore expects it to remain valid for a broader class of weight functions.
