Holomorphic minorants of plurisubharmonic functions

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Let φ be a plurisubharmonic function on a pseudoconvex domain D ⊂ Cⁿ. We show that there exists a nonzero holomorphic function f on D such that some local mean value of φ with logarithmic additional terms majorizes |f|. A similar problem is discussed for a locally integrable function on D in terms of balayage of positive measures. Key words: holomorphic function, plurisubharmonicity, minorant, balayage, Jensen inequality, mean value in the ball.