An application of the open mapping theorem

Question

Let $U\subseteq \mathbb C$ be a domain and $a,b,c \in \mathbb R$ with $a^2+b^2>0$. Determine all on $U$ holomorphic functions $f$ which satisfy:

$a\cdot Re(f) + b\cdot Im(f) +c = 0$.

I should use the open mapping theorem, which says that $f(U)$ is a domain if $U$ is a domain and $f$ a nonconstant holomorphic function. I don't know how to proceed.

Answer 1

Hint: Your hypotheses imply $f(U) \subset L,$ where $L$ is a familiar geometric object.