  Let $\varphi$ be a plurisubharmonic function on a pseudoconvex domain $D
\subset \mathbb C^n$. We show that there exists a nonzero holomorphic function
$f$ on $D$ such that some local mean value of $\varphi$ 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.
Key words: holomorphic function, plurisubharmonicity, minorant, balayage,
Jensen inequality, mean value in the ball.
