Holomorphic Minorants of Plurisubharmonic Functions

Let phi be a plurisubharmonic function on a pseudoconvex domain D in an n-dimensional complex space. We show that there exists a nonzero holomorphic function f on D such that some local mean value of phi with logarithmic additional terms majorizes log|f|. A similar problem is discussed for a locally integrable function on D in terms of balayage of positive measures.