Do you have an idea how to solve this question:
Let $f(z)$ entire function that confirm: $|f(x+iy)|\leq e^x$ for every $x,y \in \mathbb R$.
I need to prove that 1) there exist $c \in \mathbb C$ such that $|c| \leq 1$ and $f(z)=ce^z$ for every $z\in \mathbb C$.
2) what about $|f(z)| \leq e^{|z|}$ for every $z \in \mathbb C$, is 1) stay correct?
Thank You.