Let $u$ be a harmonic function on a simply connected domain $\Omega$. Prove that $u$ does not achieve maximum in $\Omega$.
What I have shown is that for any $u$ harmonic function on $\Omega$, $u(z)=\log|f(z)|$ for a function $f$ holomorphic in $\Omega$. I did this by finding the harmonic conjugate $v(z)$ of $u$ and let $f(z)= e^{u(z)+iv(z)}$.
Assume that $u$ achieves maximum in $\Omega$, then $\log|f(z)|$ achieves maximum in $\Omega$ and since $\log$ is an increasing function, $f$ achieves maximum in $\Omega$, and then I can use Maximum Modulus Principle + Identity Principle to show that $f$ is constant and hence $u$ is constant. However, I am unable to proceed here. Either the question is missing the assumption that $u$ is non-constant, or my approach is wrong.