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](https://arxiv.org/pdf/1709.03112.pdf)* (Example 1.1 and Question 3.3).



Zeros of an infinite series
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.


