[Math] If $f$ and $g$ are holomorphic, then $\log(|f|+|g|)$ is subharmonic

Question

Let $f$ and $g$ be two holomorphic functions on a plane domain, and let $u(z)=\log(|f(z)|+|g(z)|)$. Is it true in general that $u$ is subharmonic?

I know it is true if $g=0$, but here I have some doubts… Please help me!

Answer 1

Yes, this is true. More generally: a function $g$ is called log-subharmonic if its logarithm is a subharmonic function. As you know, both $|f|$ and $|g|$ are log-subharmonic. And the fact is that the sum of two log-subharmonic functions is also log-subharmonic. You can find a proof of this fact in Hypercontractivity for log-subharmonic functions by Graczyk, Kemp, Loeb, and Zak (see Proposition 2.2).

I'll sketch a proof for your particular case. Let $z_0$ be a point where $|f(z_0)|\ne 0$. Write $u$ as $\log|f| + \log(1+|g/f|) $ in a neighborhood of $z_0$. The first term is subharmonic. The second term is of the form $\log(1+e^v)$ where $v=\log |g/f|$ is subharmonic in a neighborhood of $z_0$. The function $\phi(t)= \log(1+e^t)$ is convex and increasing, as you can check by taking its derivatives. Therefore, $\phi\circ v$ is subharmonic.

If $|g(z_0)|\ne 0$, the above applies with $f,g$ interchanged. And if both $f,g$ vanish at $z_0$, then $u(z_0) = -\infty$, so the mean value inequality holds for $z_0$.