Bôcher's theorem in \(\mathbb{R}^ 2\) and Carathéodory's inequality (Q1332592)

``Let \(\varphi(z)= o(| z|^{-s})\) when \(| z|\to 0\) be a real-valued function with \(s\leq 1\). Suppose \(u(z)\) is harmonic in \(0< | z|<1\) such that \(u(z)\geq \varphi(z)\). Then \(u(z)= \lambda\log | z|+ v(z)\) where \(v(z)\) is harmonic in \(| z|<1\).'' The authors have not observed that by adding a certain term \(\alpha\log | z|\) to \(u(z)\) one obtains the real part of a holomorphic function with a possible singularity at \(z=0\). By the well-known properties of a function around such a point the result is trivial.