Equilibrium points of logarithmic potentials on convex domains

Let $D$ be a convex domain in the plane. Let $a_k$ be summable positive constants and let each $z_k$ lie in $D$. If the $z_k$ converge sufficiently rapidly to a boundary point of $D$ from within an appropriate Stolz angle then the function $f(z) = \sum_{k=1}^\infty a_k /(z - z_k)$ 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.