Past exam of the metric spaces and complex analysis course of the University of Oxford / 2023 / 6 / c / i

Suppose $F:\mathbb{D}\to \mathbb{C}$ were a holomorphic bijection. By the allowed nonvanishing derivative result, $F^{\prime }(z)\ne 0$ 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. ThereforeThis is the analytic obstruction behind the fact that the complex plane and unit disc are homeomorphic but not conformally equivalent.