Let $f$ be a non constant entire function satisfying the following two conditions:
a) $f(0)=0$
b) For every positive real number $M$, the set $\{z:|f(z)|<M\}$ is connected.
Prove that $f(z)=cz^n$ for some constant $c$ and positive integer $n$.
Since zeroes of an analytic function are isolated, I can take a closed disc of positive radius in which $0$ is the only zero of $f$. The conclusion I need, suggests that I must prove that $f$ has a pole at infinity. That will prove that $f$ is a polynomial. In addition I must prove that $0$ is the only zero for $f$. Is there any more elegant way of approaching the solution? Help me to prove the two assertions I made above.
