Zeros of an infinite series

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).

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.

• What do you mean by "the is true"? – Peter Mortensen Jan 20 at 2:49
• @Peter Mortensen: Thanks. I corrected the misprint. – Alexandre Eremenko Jan 20 at 3:53