= Solution
Suppose $F:\mathbb D\to\mathbb C$ were a <holomorphic> bijection. By the allowed nonvanishing derivative result, $F'(z)\ne0$ throughout $\mathbb D$. The <holomorphic inverse function theorem> gives a <holomorphic> inverse locally at every image point. Since $F$ is globally bijective, these local inverses agree with its global inverse $H:\mathbb C\to\mathbb D$.
Thus $H$ is an <entire function> with $|H|<1$. The <Liouville theorem> forces $H$ to be constant, contradicting its being an inverse. Therefore
$$
\boxed{\text{there is no holomorphic bijection }\mathbb D\to\mathbb C}.
$$
This is the analytic obstruction behind the fact that the <complex plane and unit disc are homeomorphic but not conformally equivalent>.
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