= Solution
If a <holomorphic> <bijection> existed, its inverse would be an entire function taking values in the <unit disc>. <Liouville theorem> would make that inverse constant, contradicting bijectivity. Thus \b[no <holomorphic> <bijection> exists].
A <continuous> <bijection> does exist, even a <homeomorphism>:
$$
\boxed{h(z)=\frac{z}{1-|z|},\qquad h^{-1}(w)=\frac{w}{1+|w|}.}
$$
Both maps preserve the argument and change the radius by the mutually inverse strictly increasing functions $r\mapsto r/(1-r)$ and $s\mapsto s/(1+s)$. The displayed formulas also prove <continuity> at zero and everywhere else.
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