Equilibrium points of logarithmic potentials on convex domains

Let $D$ be a convex domain in $\mathbb{C}$. Let $a_k > 0$ be summable constants and let $z_k \in D$. If the $z_k$ converge sufficiently rapidly to $\zeta \in \partial D$ from within an appropriate Stolz angle, then the function $\sum_{k=1}^\infty a_k /( z - z_k )$ has infinitely many zeros in $D$. An example shows that the hypotheses on the $z_k$ are not redundant and that two recently advanced conjectures are false.