On the generalization of Frostman's theorem due to S. Kobayashi (Q1114039)

For a single-valued meromorphic function f(z) in a domain D of the z- plane and a boundary point \(\zeta\) of D, the range of values \[ R_ D(f,\zeta)=\cap_{r>0}f(D\cap U(\zeta,r)), \] where U(\(\zeta\),r) denotes the open disk \(| z-\zeta | <r\), and the least harmonic majorant of \(| f(z)|\) is denoted by \(H_{| f|}(z)\). In this paper, the author proves the following generalization of Frostman's theorem. Theorem. Suppose that \(| f(z)| <1\) in D and that there exists a sequence \(\{z_ n\}\) of points in D converging to \(\zeta\in \partial D\) for which \(H_{| f|}(z_ n)\to 1\) and \(f(z_ n)\to a\), with \(| a| <1\) as \(n\to \infty\). Then we have the following alternatives: (1) the range of values \(R_ D(f,\zeta)\) covers the unit disk except possibly for a set of logarithmic capacity zero; this is always the case if \(\zeta\) is a regular boundary point with respect to the Dirichlet problem. (2) \(H_{| f|}(z)\equiv 1\) in D and there exists \(r_ 0>0\) such that \(\partial D\cap \overline{U(\zeta,r_ 0)}\) is of capacity zero, so that f is analytic throughout in \(U(\zeta,r_ 0).\) The assumption in the theorem above is a little weaker than that of \textit{S. Kobayashi} [J. Anal. Math. 49, 203-211 (1987; Zbl 0644.30079)].