Martin boundaries of plane domains (Q1073214)

The most part of this paper is devoted to a proof of the following result: If \(\Omega\) is a plane domain of the form \(\Omega =\{| z| <1\}-F\) with a closed set F in \(\{\) Re \(z\leq 0\}\) containing 0, then for any \(0<\delta <\pi /2\) the closure of the sector \(\{\) \(| \arg z| <(\pi /2)-\delta\), \(| z| <r\}\) \((r<1)\) in the Martin compactification of \(\Omega\) contains only one Martin boundary point, which is minimal. This is a special case of a result obtained by \textit{A. Ancona} [Ann. Inst. Fourier 29, No.4, 71-90 (1979)]. Another type of plane domains \(\Omega\) having only one minimal N-Martin (i.e., Kuramochi) boundary point over the origin is also given.