Let $\mathcal{f}$:$\mathrm{D}$$\to$$\mathbb{C}$, $\mathrm{D}$={$\mathcal{z}$ $\in$ $\mathbb{C}$ : $\vert$$\mathcal{z}$$\vert$ < 1} be a holomorphic, bounded, non-constant function. Show that for the roots ($\mathcal{z_k)_{k=1}^N}$ ($\mathcal{N}$ $\in$ $\mathbb{N}$ $\cup$ $\infty$)
$\sum_{n=1}^\mathcal{N} (1-\vert$$\mathcal{z_n}$$\vert$) < $\infty$
holds
I know that I need to use Jensen's formula, but it is not clear me that $\mathrm{D}$ contains the unit closed disc, which required for the use of Jensen'formula. Then, if $\mathcal{f}$ is bounded on $\mathrm{D}$, I assume
log($\vert$$\mathcal{f(0)}$$\vert$) is also bounded, so is $\sum_{k=1}^N log(\vert$$\mathcal{z_k}$$\vert$), then also $\prod_{k=1}^N$$\vert$$\mathcal{z_k}$$\vert$, and with with a contradiction, I think I can get the final result. But is this correct if $\mathcal{N}$=$\infty$ ?