= Liouville rigidity from a bounded holomorphic embedding
Suppose $D\subseteq\mathbb C$ is <open> and admits an <injective> <holomorphic> map $\Phi:D\to\mathbb C$ whose <image> is <bounded>. Every <entire function> $f:\mathbb C\to D$ is constant: the <function composition> $\Phi\circ f$ is a <bounded> <entire function>, hence constant by <Liouville theorem>, and <injectivity> of $\Phi$ then forces $f$ to be constant. Surjectivity of $\Phi$ onto a disk is unnecessary. For example $z\mapsto z/(1-z)$ is a <conformal map> from $\mathbb C\setminus[0,1]$ onto $\bigl(\mathbb C\setminus[0,\infty)\bigr)\setminus\{-1\}$. A square-root branch with argument in $(0,2\pi)$ followed by the <Cayley transform between the half-plane and disk> maps this to the punctured unit disk. It therefore supplies a <bounded> <injective> <holomorphic> coordinate on the interval complement, giving the same rigidity even though that domain is not simply connected.
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