User:DVD206/Blaschke products

Let a_i be a set of n points in the complex unit disc D. The corresponding Blaschke product is defined as

$B(z)=\prod _{k}{\frac {|a_{k}|}{a_{k}}}({\frac {a_{k}-z}{1-{\overline {a_{k}}}z}}).$

If the set of points is finite, the function defines the n-to-1 map of the unit disc onto itself,

$f:\mathbb {D} {\xrightarrow[{}]{n\leftrightarrow 1}}\mathbb {D} .$

If the set of points is infinite, the product converges and defines an automorphism of the complex unit disc, given the Blaschke condition

$\sum _{k}(1-|a_{k}|)<\infty .$

The following fact will be useful in our calculations:

$B(0)=\prod _{n}{|a_{i}|}.$