Complex plane is not conformally equivalent to the upper half-plane
= Complex plane is not conformally equivalent to the upper half-plane
Composing a hypothetical conformal equivalence $\mathbb C\to\mathbb H$ with a <Cayley transform (hyperbolic geometry)> $\mathbb H\to\mathbb D$ would give a nonconstant bounded <entire function>, contradicting the <Liouville theorem>.