Critical Points of Quotients of Finite Blaschke Products of Small
Degrees

Blaschke products are a class of inner functions that map the unit disk onto itself and the unit circle onto itself, preserving the boundary modulus equal to one, and they play a fundamental role in complex analysis and geometric properties.
The quotients of finite Blaschke products do not map the unit disk onto itself, since they may have poles inside the unit disk, but they preserve the unit circle, maintaining unimodular boundary values almost everywhere on it.
The present study systematically investigates the critical points of quotients of Blaschke products of degrees two and three,
$B(z) = z \left(\displaystyle\frac{z-a}{1-\bar{a}z}\right), |a|>1$,
and there are two distinct critical points on the unit circle given by the intersection of the circles $|z-a|=\sqrt{|a|^{2}-1} ~\text{and}~ |z|=1$. For the case of degree three,
$B(z)=z\left( \displaystyle\frac{z-a}{1-\bar{a}z} \right)\left( \displaystyle\frac{z-b}{1-\bar{b}z} \right)$,
where $a,b > 1$, the critical points are determined by a degree four polynomial. This coincidence shows that the quotient of Blaschke products always has at least one critical point on the unit circle.
These results establish a connection between the critical points and the locus of points, satisfying the pre-image of the unit circle under finite Blaschke products of degrees two and three, as well as under quotients of Blaschke products.