Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 3G / Solution

If such an $f$ existed, boundedness would make its isolated singularity at zero removable. Let $F$ be the holomorphic extension to the disc. Continuity gives $1\le |F(0)|\le 2$. Equality at either endpoint contradicts the minimum-modulus principle applied to $F$ or the maximum modulus principle, since $F$ is nonconstant. Hence $F(0)\in A$. Surjectivity of $f$ supplies some nonzero $z$ with $f(z)=F(0)$, contradicting injectivity of the extension. Therefore