Proof that two conditions imply that a function $f$ is constant

Question

1. Let $\Omega \subset \mathbb{C}$ open. How do I prove that if $f(\Omega) \subseteq \text{ a line }$ then $f$ is constant?
2. How do I prove that if $f$ is holomorphic in $\mathbb{C}$ and there exists $r>0$ such that $f(\mathbb{C})\subset \mathbb{C}-B(0,r)$, then $f$ is constant?

Answer 1

1. Since $f$ is holomorphic it maps open sets to open sets,since $f(\Omega)\subset \text{line}\implies f \text{is constant}$
2. Use Picards Theorem ,If the range of $f$ excludes more than two points of $\Bbb C$ then $f$ is constant.Here it excludes uncountably many points.

Answer 2

A proof of 2. without Picard:

We have $|f(z)| \ge r$ for all $z \in \mathbb C$. Let $g=1/f$. Then $g$ is a bounded entire function. By Liouville, $g$ is constant, hence $f$ is constant.