A Blaschke product is a function of the form $$B(z):=z^k\prod_{n=1}^{\infty}\frac{a_n-z}{1-\overline{a_n}z}\frac{|a_n|}{a_n}$$ where the $a_n$ are the non-zero zeros of $B$, and satisfie $\sum_{n=1}^{\infty}(1-|a_n|) < \infty$.
Blashke products are holomorphic and bounded by 1 on the unit disk. A well known theorem asserts that $B$ has radial limits almost everywhere on the unit circle, i.e. that the limit $$\lim_{r \rightarrow 1} B(re^{i \theta})$$ exist for almost every $\theta$. I'm looking for an example of Blashke product such that the radial limit does not exist at a certain point, say $1$ for example. In particular, a Blaschke product with zeros in $(0,1)$ such that $$\limsup_{r \rightarrow 1}|B(r)| =1$$ would work.
Does anyone have a construction or reference?
Thank you, Malik