The question is the following:
Let $a \in \mathbb{R}$ and $f$, $g$ entire functions such that $$Re(f(z)) \leq a Re(g(z))$$ for every $z \in \mathbb{C}$. prove that exists $c \in \mathbb{C}$ such that $f(z) =a g(z)+c$
I'm familiarized with Liouville Theorem, maxim modulus principle and this kind of stuff related with Local Cahuchy properties. My guess was to try to prove that f and g were bounded and then apply Lioville Theorem to say that they were constant and then by using the inequality show that f must be $ag(z)$ $+$ $c$ . However, I didn't manage to find a way to do that and I'm not even sure if that's the correct way of solving it.
I would be very gratefull if someone could help me. Thanks in advance!
Let $\varphi=e^{f-ag}$, then $$ |\varphi|=e^{\text{Re}\,f-a\text{Re}\,g}\leqslant 1 $$ Since $f$ and $g$ are entire functions, $\varphi$ is also an entire function and thus according to Liouville theorem, $\varphi$ is constant. Thus there exists $k:\mathbb{C}\rightarrow\mathbb{Z}$ such that $f-ag=2i\pi k$ and since $k$ is continuous, $k$ is constant according to the intermediate value theorem. Finally there exists $c\in\mathbb{C}$ such that $f=ag+c$.