= Blaschke condition
{c}
{title2=$\sum_n(1-|a_n|)<\infty$}
The <Blaschke condition> for a sequence in the <unit disc> is $\sum_n(1-|a_n|)<\infty$. It characterizes the zero sequences, counted with <multiplicity>, of nonzero bounded <holomorphic functions>. Sufficiency follows by constructing the <Blaschke product>. For necessity, <Jensen's formula> gives $\sum_{|a_n|<r}\log(r/|a_n|)\le\log\|f\|_\infty-\log|f(0)|$ when $f(0)\ne0$. Letting $r\uparrow1$ and using $-\log|a|\ge1-|a|$ proves the condition. A finite zero at zero is factored out first. Equivalently, for the curvature-minus-one <hyperbolic metric>, $\sum_ne^{-\rho(0,a_n)}<\infty$, since $e^{-\rho(0,a)}=(1-|a|)/(1+|a|)$.
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