Prove that $\log(f+g)$ be a plurisubharmonic function

Question

Suppose $G$ is an open set of $E$,($E$ is complex Banach space) and $f,~g :G \to \left[0,\infty \right)$ such that $\log f$ and $\log g$ be two plurisubharmonic(PSH) functions in $G$.

Prove that $\log(f+g) \in \text{PSH}(G)$.

I have tried...but... :(.

Answer 1

You probably know (otherwise it is not hard to prove) that

If $\phi : \mathbb{R}^n \to \mathbb{R}$ is increasing in each variable and convex, and $u_1, \ldots, u_n$ are plurisubharmonic, then $\phi(u_1, \ldots, u_n)$ is also purisubharmonic.

For your situation, take $\phi(x,y) = \log(e^x + e^y)$.