Let $\Omega \subseteq \Bbb C$ be an open and connected set and let $f:\Omega \to \Bbb C$ be an analytic function .Pick out true ones:

1. $f$ is bounded if $\Omega $ is bounded

2.$f$ is bounded only if $\Omega $ is bounded

3.$f$ is bounded iff $\Omega $ is bounded.

For $3$ I took $\Omega=\Bbb C$ and I took the map $z\mapsto e^{iy}$ where $z=x+iy $. Then $|f(z)|=1$ which is bounded but $\Omega=\Bbb C$ is not.

Am I right? For $1,2$ I am not getting any examples .Any help will behelpful