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



Minimum of two plurisubharmonic functions
Here is a counterexample to your statement for $n=1$. Let $D=\{ x+iy:|x|<1,|y|<\pi/2\}$.
Functions $u\_1(x,y)=e^x\cos y$ and $u\_2(x,y)=e^{-x}\cos y$ are harmonic. Their minimum
is $e^{-|x|}\cos y$ which is equal to $1$ at $0$, and the maximum on $\partial D$ is
$e^{-1}<1$. It remains to notice that $\exp(u\_1)=|\exp\exp z|$ and $\exp(u\_2)=|\exp\exp(-z)|,$ so for $f(z)=\exp\exp(z)$ and $g(z)=\exp\exp(-z)$ your statement does not hold. 


