# Inequality of an entire function using liouville

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.

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 $$h \in \mathbb{C}$$.