ABSTRACT
Plurisubharmonic functions have proved quite useful in the further investigation of domains of holomorphy and related topics, following the pioneering work of K. Oka. Recall from section G that an open subset D ⊆ ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq1913.tif"/> is a domain of holomorphy precisely when it is holomorphically convex, and that the latter condition is expressed in terms of functions of the form |f| where f is holomorphic in D. Now it is a familiar and obvious consequence of the Cauchy integral formula that the absolute value u = |f| of a holomorphic function f of a single variable satisfies the integral inequality J(2) and hence is a subharmonic function. And since the restriction of a holomorphic function of several variables to any complex line is a holomorphic function of one variable, it is evident that the absolute value |f| of a holomorphic function f of several variables is a plurisubharmonic function. That suggests looking at corresponding notions of convexity of domains defined in terms of plurisubharmonic functions, as a fairly natural generalization of holomorphic convexity. This suggestion turns out to be very fruitful indeed, as will become apparent from the subsequent discussion. Perhaps the easiest place to begin such an investigation is with the following.
