Subharmonic functions Consider:

if $\log u$ and $\log v$ are both subharmonic function on an open set $U$, then $\log(u+v)$ is also subharmonic on $U$.

How can I prove this?

If $w$ is strictly positive and $C^2$ then a direct calculation shows that $\log w$ is subharmonic if and only if $w\Delta w+|\nabla w|^2 \ge 0$.  Taking $w=u+v$ we see that $\log(u+v)$ is subharmonic if and only if
$$
(u\Delta u+|\nabla u|^2)+(v\Delta v+|\nabla v|^2) +\Delta(uv) \ge 0.
$$
If $\log u$ and $\log v$ are subharmonic then  the first two terms on the left are non-negative. And so is the third because 
$$
\Delta(uv) = uv\cdot\left[\Delta(\log u+\log v) +|\nabla(\log uv|^2\right],
$$
which is clearly non-negative.
