Equilibrium points of logarithmic potentials on convex domains

الأصل · EN

Let D be a convex domain in the plane. Let aₖ be summable positive constants and let each zₖ lie in D. If the zₖ converge sufficiently rapidly to a boundary point of D from within an appropriate Stolz angle then the function f(z) = ∑ₖ₌₁∞ aₖ /(z - zₖ) has infinitely many zeros in D. An example shows that the hypotheses on the zₖ are not redundant, and that two recently advanced conjectures are false.