A property of holomorphic functions related to plurisubharmonic functions

Let $\phi$ be a negative plurisubharmonic function in a pseudoconvex domain $\mathrm{\Omega }$ in ${\mathbb{C}}^n$ and $f$ be a bounded holomorphic function belonging to $L^2(\mathrm{\Omega },\phi )$. For all negative plurisubharmonic functions $\psi$ satisfying that $\psi$ is close to $\phi$ in the $L^1(\mathrm{\Omega })$ norm. Using the recent results about $L^2-$estimate for $\bar{\mathrm{\partial }}-$equation and strong openness conjecture, we prove that there exists a holomorphic function $g$ belonging to $L^2(\mathrm{\Omega },\psi )$ such that $g$ is close to $f$ in the $L^2$ norm on some relatively compact open subset ${\mathrm{\Omega }}^{{}^{\mathrm{\prime }}}$ of $\mathrm{\Omega }\mathrm{.}$