ABSTRACT
Plurisubharmonic functions played an important role in the preceding discussion of domains of holomorphy, and they have many more relations to and consequences for holomorphic functions, just two of which will be discussed briefly here. One of these is a useful extension of Riemann’s removable singularities theorem. It was noted at the beginning of section M that |f| is plurisubharmonic whenever f is holomorphic. Actually it follows immediately from Jensen’s inequality in several variables, Theorem A8, that log |f| is plurisubharmonic whenever f is holomorphic. This is a somewhat stronger result, since the real exponential function is convex and monotonically increasing, so by Theorem K5(d) the plurisubharmonicity of log |f| implies that of |f| = exp log |f|. The zero locus of f is the set of points at which the function log |f| takes the value −∞. The major role played by the zero sets of holomorphic functions thus suggests introducing the following notion.
