Find all entire functions with $|f(z)|\geq e^{|z|}$ for all $z \in \mathbb{C}$. Find all entire functions with $|f(z)|\geq e^{|z|}$ for all $z \in \mathbb{C}$. 
I don't think there is any such entire function, and here is my thought: since $\Re(z) \leq |z|$, we know $|e^z|\leq  e^{|z|}\leq |f(z)|$ for all $z$. Consider $g(z)=\frac{f(z)}{e^z}$. Since $e^z$ is never $0$, $g(z)$ is an entire function. But $|g(z)| \geq 1$ for all $z\in \mathbb{C}$, so there is a contradiction. 
I was wondering is there any hole in the preceding argument? 

It's a bit more complicated than necessary.  $|f(z)| \ge e^{|z|} \ge 1$, so  why not just use $f$ instead of $g$?  
And, by the way, what are you contradicting?
