A couple more questions about entire functions that I'm having difficulty with:
(1) Suppose $f$ is entire with $f(0)=0$ and $|f(z)|\leq e^{1/|z|}$ for all $z\neq0$. Must $f$ be identically $0$?
(2) Suppose that $g$ is entire with $g\circ g=g$. If $g$ is not constant, must $g$ be the identity?
Thanks again for any/all advice.