I need help with the following problem:
Let $f: B_1(0) \to \mathbb C$, where $B_1(0)$ denotes the unit disk, be a holomorphic, bounded and non-constant function with zeros $(z_k)_{k\in \mathbb N}$ (in particular, $\lvert z_k \rvert < 1$ for all $k\in \mathbb N$. Prove: $$\sum_{n=1}^\infty (1-\lvert z_k \rvert) < \infty$$ Hint: Use Jensen's formula.
My thoughts:
Obviously, we need $\lvert z_k \rvert \to 1$ for that sum to converge.
This is already were I struggle. Why does every function of this type have infinitely many zeros near $|z|=1$ and impossibly at any other arbitrary value $|z| < 1$? I don't really have a good intuition here. Also this is required to use Jensen's formula.
The other thing I tried to show is $f(0) \neq 0$. Why is that? I don't see why it shouldn't be possible to have a function with the given properties and with $f(0) = 0$.
By hypothesis, there are infinitely many $z_i$. You can arrange them so that $|z_1|\le|z_2|\le\cdots$. If $|z_n|\not\to 1$ then $|z_n|\to a<1$. The closed disc $\overline{B_a(0)}$ would have infinitely many zeros of $f$, which is impossible by a compactness argument. So $|z_n|\to1$.
Jensen's formula usually assumes $f(0)\ne0$, but if $f(0)=0$ write $f(z)=z^k g(z)$ with $g(0)=0$ and apply it to $g$ instead.