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How can one show that

if $f$ is entire and $f(\mathbb{C})\subset \mathbb{C} \setminus p$ where $p$ is any ray in $\mathbb{C}$ then $f$ is constant.

It's obvious result from the Little Picard Theorem.

But how can one prove it using only Liouville Theorem?

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Hint: Map bijectively $\mathbb{C} \backslash p$ to a half plane ( $z \mapsto \sqrt{z}$) and then to a disk.

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