TITLE: More Properties of Entire Functions
QUESTION [6 upvotes]: 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.

REPLY [13 votes]: (1) If $f$ is bounded on $\{z:|z|\leq M\}$ and on $\{z:|z|\geq M\}$, then $f$ is bounded on $\mathbb{C}$.
(2) The equation $g\circ g=g$ implies that $g$ is the identity when restricted to the range of $g$.  The range of a nonconstant analytic function is always so big that if 2 analytic functions agree on this range, then they must agree everywhere.  (E.g., because the range is uncountable, or because it is open.)