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Let $\Omega$ be a pseudo convex domain. Let $r$ be any $C^2$ function $r: \mathbb C^2\to \mathbb R$. Let $\Omega: \{z: r(z)<0\}$. Then we know that $\psi: -log(-r)+\lambda |z|^2$ is a plurisubharmonic function for large $\lambda$.

Is $-log(-r)$ also plurisubharmonic? May I have an answer or even a hint? Thanks in advance.

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Pseudoconvexity of $\Omega$ imposes no restrictions on the function $r$ in the domain $\Omega'=\{z: r<-1\}$, say. Thus, you can modify a given psh function $r$ to have any local behavior in $\Omega'$ (as long as its $C^0$-norm does not change much). This will destroy psh property and will give you a counter-example. – Misha Jun 6 '12 at 12:27

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