Unbounded Bloch function I'm studying the Bloch space $\mathcal{B}$; this is the set of all analytic fuctions $f$ defined on the open unit disc $\mathbb{D}$ such that $$\sup_{|z|<1}(1-|z|^2)|f'(z)|<\infty$$ My question is more curious: can you give an example of an unbounded Bloch function?

$f(z) = \mathrm{argtanh}\,z$ is analytic with $|f(z)| \to +\infty$ as $z\to 1$. Since $f'(z) = \frac{1}{1-z^2}$, it satisfies $(1-|z|^2)|f'(z)| \leqslant 1$.
