Equilibrium points of logarithmic potentials on convex domains

Abstract

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