= Blaschke product
{c}
{title2=$B(z)=\lambda z^m\prod_n b_{a_n}(z)$}
{wiki}
For zeros $a_n$ in the <unit disc> satisfying $\sum_n(1-|a_n|)<\infty$, with $m$ occurrences at zero, a <Blaschke product> is
$$
B(z)=\lambda z^m\prod_{a_n\ne0}\frac{|a_n|}{a_n}\frac{a_n-z}{1-\overline{a_n}z},\qquad |\lambda|=1.
$$
For $|z|\le R<1$ and $a\ne0$, the normalized factor $b_a$ satisfies $|1-b_a(z)|\le(1-|a|)(1+R)/(1-R)$. The product therefore converges locally uniformly; outside its prescribed zeros the logarithms of its tail factors converge absolutely, so it has no additional zeros. Every finite product has modulus at most one in the disc, and the limit does too. Since $|b_a(w)|$ is <pseudohyperbolic distance> from $w$ to $a$, its modulus can be estimated geometrically.
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