I am trying to prove the following conjecture:
Let $f(z)$ and $g(z)$ be holomorphic functions defined on a simplify connected subregion $\Omega$ of the complex plane, where $\forall z\in \Omega$, $f(z)\neq 0$.
Prove that the function $|f|-|g|$ attains its minimum on the boundary of $\Omega$.
Since $f$ and $g$ are holomorphic, their modulus $|f|$ and $|g|$ are subharmonic and attains maximum on the boundary. $-|g|$ is superharmonic and attains minimum on the boundary. Since $f$ does not vanish in the domain $log|f|$ is harmonic and attains minimum on the boundary. So in fact since $log$ is a monotonic function, $|f|$ also attains minimum on the boundary.
In total we have a sum of two functions, $|f|$ and $-|g|$, both attain their minimum on the boundary. For general functions, this does not imply that the sum should attain minimum on the boundary. However, I believe that it is true in this particular case.
Any help proving this (or finding a counter example) would be appreciated!