 Blaschke product

For zeros $a_n$ in the unit disc satisfying ${\sum }_n(1-|a_n|)<\infty$, with $m$ occurrences at zero, a Blaschke product isFor $|z|\le R<1$ and $a\ne 0$, 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.

A Blaschke product is a specific type of function in complex analysis that is defined as a product of terms related to the holomorphic function behavior on the unit disk. Specifically, a Blaschke product is constructed using zeros that lie inside the unit disk. It is a powerful tool in the study of operator theory and function theory on the unit disk. Formally, if \(\{a_n\}\) is a sequence of points inside the unit disk (i.e.