One of the questions in my complex analysis book (Stein's text) is the following:
Prove that if $f$ is holomorphic in the unit disc, bounded, and not identically zero, and $z_1,z_2,\ldots,z_n,\ldots$ are its zeros, $(|z_k|<1)$, then
$$
\sum_{n=1}^\infty(1-|z_n|)<\infty.
$$
I proved this just fine using Jensen's formula, but I am still not able to think of an example for such a function. Obviously it will have to have infinitely many zeros, otherwise we're only adding finitely many terms and the problem becomes trivial. Since there are infinitely many, the limit point(s) has to be on the boundary (otherwise the function is identically zero). At one point, I think someone suggested a function like $\sin(\pi/z)$ but this isn't bounded (indeed, this has lots of problems around $0$).
Does anyone have an example of such a function?
Best Answer
The simplest examples are probably infinite Blaschke products:
$$f(z) = \prod_{n=1}^\infty \frac{z-z_n}{1-\bar z_n z}.$$
You can check that this product converges when your condition (which is often called the Blaschke condition) is satsified.