In Steven Krantz's book on several complex variables he has the following problem: Suppose $\log u_1$ and $\log u_2$ are subharmonic on an open subset of the complex plane. Show that $\log (u_1+u_2)$ is also subharmonic.
I attempted to do this first by looking at averages, but couldn't get a tight enough inequality. Then I attempted to look at the Laplacian, but got stuck in the $1$d case showing that $f''g+fg''>2f'g'$ if $\log f$ and $\log g$ are subharmonic. Any hints on any of this would be wonderful.
Let $\Omega\subseteq \mathbb C$ be the open set.
First, you want to prove that $\log(u_1+u_2)$ is upper semicontinuous on $\Omega$.
$\underline{\text{Hint}}$ : Prove that $u_1+u_2$ is subharmonic.
Then, take $x$ in $\Omega$ and $r>0$ such that $\bar{B}(x,r)\subseteq\Omega$.
Let $h$ be a real-valued continuous function on $\bar{B}(x,r)$ such that $h$ is harmonic on $B(x,r)$ and $\log(u_1+u_2)\leqslant h$ on $\partial B(x,r)$. You want to prove that this inequality holds on $B(x,r)$.
$\underline{\text{Hint}}$ : Take the exponential on each side of the inequality, and prove that $u_k e^{-h}$ is subharmonic on $\Omega$ for $k\in\{1,2\}$