A linear holomorphic function

Question

If $f$ is a holomorphic function on a simply connected open domain $\Omega$, and $f$ is a linear function on the boundary $\Omega$, i.e. $f=az+b$ on $\partial\Omega$. Then, can I say that $f$ is also linear function on $\Omega$?

Answer 1

Since $f(z)=az+b$ on the boundary, we know $g(z):=f(z)-az-b=0$ on $\partial \Omega$. By the maximum modulus principle, $g(z)=0$ for all $z\in\Omega$, or equivalently, $f(z)=az+b$.