On local harmonic majorants of subharmonic functions (Q1239290)

In this note the author proves that a function \(f\), subharmonic in the interior \(S\) of a compact bordered Riemann surface, which admits a local harmonic majorant at each point of the boundary \(B\) of the surface, must then have a harmonic majorant in the interior \(S\). The author's proof of this result makes use of the proof by \textit{P. M.Gauthier} and \textit{W. Hengartner} [J. Analyse math. 26, 405--412 (1973; Zbl 0079.31001)] of that same result where \(S\) is the unit disc. The author offers a second proof of the result for the special case that is more elementary than the first one. Then the author uses the second proof to show that a function \(F\) meromorphic in \(S\) above \(S\) of bounded characteristic there if and only if \(F\) is of bounded characteristic at each point of the boundary \(B\) (i.e., \(F\) is locally of bounded characteristic). This last result is essentially due to \textit{M. Heins} [J. Analyse math. 18, 121--131 (1967; Zbl 0152.06902)] but the present proof seems to be a much more elementary one than that due to Heins.