Let $f$ be an entire function and $L$ a line in $\mathbb{C}$ such that $f(\mathbb{C})\cap L=\emptyset$. Show that $f$ is constant function.

Hint Show that $f(\mathbb{C})$ is (path) connected.

Since $f(\mathbb C)$ doesn't intersect the line, it is included in one of the half planes defined by $L$.

Next, pick some $w$ in the other half plane, and show that $$g(z)=\frac{1}{f(z)-w}$$ is entire and bounded.

Wlog. $L$ is the $x$-axis and $f(\Bbb C)\subseteq \Bbb H$. Note that you can map $\Bbb H$ to $\Bbb D$.

As $f$ is entire and non-constant, $f$ takes all values with atmost one exception possibly (Little Picard's theorem). But $f(\mathbb C)\cap L=\emptyset$ implies $f$ skip infinite points on $\mathbb C$, so $f$ must be constant.