Function $f(z)$ such that $|f(z)| > |z|$ for every $z\in\mathbb{C}, 1/f(z)$
Solution 1:
If $f$ is continuous, ${1\over {|f(z)|}}<1$ for $|z|>1$ since $B(0,1)$ is compact, ${1\over {|f(z)|}}$ is bounded on $B(0,1)$. This implies that ${1\over {|f(z)|}}$ is bounded.