Zeros of an infinite series

Question

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).

Answer 1

This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:

MR2317957
Langley, J. K. Equilibrium points of logarithmic potentials on convex domains, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.

His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.

However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.

Notice that a similar question in the plane (under the assumptions $z_k\to\infty$, and $$\sum_k a_k/|z_k|<\infty,$$ $f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.