Holomorphic minorants of plurisubharmonic functions

TL;DR

This paper demonstrates the existence of nonzero holomorphic functions that serve as minorants for plurisubharmonic functions on pseudoconvex domains, utilizing mean value properties and balayage techniques.

Contribution

It establishes the existence of holomorphic minorants for plurisubharmonic functions using mean value inequalities and balayage, extending classical results in complex analysis.

Findings

Existence of holomorphic functions majorizing local mean values of plurisubharmonic functions.

Extension of minorant results to functions with measures via balayage.

Application of Jensen inequality in the context of plurisubharmonic functions.

Abstract

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.