Prove that |f|-|g| attains minimum on the boundary 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 $\Omega$.
Some insights:
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!
 A: The problem of your question can be solved by the Dirichlet principle. First I add some missing prerequisites concerning the boundary:
Claim: Let $\Omega \subset \mathbb C$ be a bounded domain where the Dirichlet principle has a solution, e.g., each component of the complement $\mathbb C - \Omega$ comprises more than one point. Consider two functions $f,g$ which are continuous on $\overline{\Omega}$ and holomorphic on $\Omega$. Assume $f$ without zeros in $\overline{\Omega}$. Then $|f|-|g|$ attends its minimum on the boundary.
Proof: The claim is equivalent to the claim that $log|f|-|g|$ attends its minimum on the boundary, because the function $log$ is strictly monotonic increasing. The latter claim is equivalent to the claim that $|g|-log|f|$ attends its maximum on the boundary.
By the Dirichlet principle a function $h$ on $\overline{\Omega}$ exists which is harmonic on $\Omega$ and equals $|g|-log|f|$ on $\partial \Omega$. By the maximum principle the harmonic function $h$ attends its maximum on $\partial \Omega$. Furthermore, the function $|g|$ is subharmonic and the function $log|f|$ is harmonic. Hence also $-log|f|$ is harmonic, specifically subharmonic, and the sum 
$$|g|+(-log|f|) = |g|-log|f|$$ 
is subharmonic. Hence for all $z \in \Omega$:
$$|g(z)|-log|f(z)| \leq h(z) \leq max\{|g(w)|-log|f(w)|: w \in \partial \Omega\}, q.e.d.$$   
