Let $\sum_{j=1}^{\infty}a_{j}$ be a convergent series of positive numbers and $\{z_{j}\}_{j=1}^\infty$ a closed discrete subset of the open unit disc $\mathbb{D}$. Then $h(z):=\sum_{j=1}^{\infty}\frac{a_{j}}{z-z_{j}}$ is a meromorphic function on $\mathbb{D}$.

If we only consider the case of infinite sum, does $h(z)=\sum_{j=1}^{\infty}\frac{a_{j}}{z-z_{j}}$ always have infinitely many zeros on $\mathbb{D}$? Note that $h$ never vanishes outside $\mathbb{D}$.

This question comes from the paper *Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities* (Example 1.1 and Question 3.3).