It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a finite Blaschke product up to some unimodular complex constant. I am trying to understand the following set: $$S=\{f:\mathbb{D}\to\mathbb{C}:f\,\text{is a finite blaschke product and}\,f(0)=f(\frac{1}{\sqrt{2}})\}$$
It is clear that every finite Blaschke product that contains the factor $z\frac{z-\frac{1}{\sqrt{2}}}{1-\frac{1}{\sqrt{2}}z}$ is in the set, and each $f$ in the set has to have atleast two zeros (otherwise it is one to one on the disk and cannot satisfy the last condition). However, given a fixed point $w_1\in\mathbb{D}$, can I always find a function $f\in S$ and a point in the disk $w_2$ such that $f$ vanishes only at $w_1,w_2$ (both are not $0$ or $\frac{1}{\sqrt{2}}$)? Moreover, is this $w_2$ unique in some sense?
This is obviously equivalent to finding $w_2$ such that $w_1w_2=(\frac{1-\sqrt{2}w_1}{\sqrt{2}-\overline{w_1}})(\frac{1-\sqrt{2}w_2}{\sqrt{2}-\overline{w_2}})$ (the RHS is plugging $\frac{1}{\sqrt{2}}$ into the Blaschke product corresponding to $w_1,w_2$ and LHS is plugging $0$ into it). I wasn't able to solve thie equation algebraically, nor show such $w_2$ exists. Any hint/help/reference would be helpful and appreciated.
