Inequality of an entire function using liouville

Question

I'm doing a complex analysis course and I have found an exercise which I can not solve. It is the following:

Let $f$ be an entire function such that $|f'(z)|\leq Ce^{\text{Re}(z)}$ with $C \ge 0$ constant for all $z \in \mathbb{C}$. What can we say about $f$?

It seems that the question is quite general, but I figure that what is asked is to find the general form of $f$.

I'm familiarized with Liouville's theorem and this kind of stuff. My guess was to use Lioville theorem because $f$ is entire and maybe also to divide $f'$ by $e^z$ which I can because $e^z \neq 0$ for all $z$. But honestly, I do not see how this can help me to solve it or even if I'm in the correct way. I would be really grateful if someone could help me.

Thanks in advance!

Answer 1

Define $g(z)=\frac{f'(z)}{e^z}$ which is by hypotesis bounded by $C$ and entire (note that $|e^z|=e^{\Re z}$).

Thus by Liouville's theorem you can conclude $g$ is costant.

Thus you have that there exists $k \in \mathbb{C}$ such that $f'(z)=ke^z$ and $|k| \leq C$.

Finally you have $f(z)=ke^z+h$ where $h \in \mathbb{C}$.