A classical question in symmetrization [cf. W. K. Hayman, Research problems in function theory, Athlone Press, London (1967; Zbl 0158.06301)] can be phrased as: let \(g\) be the Green function for \(U\) with pole at \(w\in U\). Then do we have \[ g(re^{i\theta},|w|;U^*)\geq \widetilde{g} (re^{i\theta},w;U)? \tag{1} \] In (1) \(U^*\) is the circular symmetrization of \(U\) (so \(U^*= \{re^{i\theta}\); \(\theta\leq \theta(r)\}\) with \(\theta(r)\) chosen so that the angular measure of \(U^*\cap \{|z|=r\}\) is that of \(U\cap \{|z|=r\}\) and \(\widetilde{g}\) is the circularly symmetric nonincreasing rearrangement of the function \(e^{i\theta}\to (g(re^{i\theta}),w;U)\).
Evidence for (1) comes from the fact, due to A. Baernstein, that (1) holds when integrated over any interval \(0\leq \theta\leq \theta_0\). However, the author presents three counterexamples to (1), together with attractive sketches of the (rather simple) corresponding regions. They are either the disk or the intersection of a disk with a half-plane with at most two simple slits removed. One of the proofs requires a computation with Maple, and the other use representations of Green functions and elementary facts about normal derivatives at the boundary. The paper is unusually well-presented.