I know that in general for $u,v\in PSH$ (plurisubharmonic) $\min\{u,v\}$ is not a $PSH$ function. Are there any known results under which conditions on $u$,$v$ a function $\min\{u,v\}$ is $PSH$? I know about the Kiselman's minimum principle, and some work of Poletsky on the disc envelope...

The problem that I have is the following: Suppose $D\subset\mathbb{C}^n$ is a Stein domain and $f$, $g$ are continuous on $\overline{D}$ and holomorphic on $D$.

Is it true that for $\Phi(z)=\min\{ ||f(z)||, ||g(z)||\}$, we have $\Phi|_D\lneq\sup_D\Phi$? (i.e. is global maximum attained only on $\partial D$) (under what conditions)

If you have any good reference I will be more than happy to look at it. Thank you

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