I'm trying to solve this question:
Let $f$ be analytic in a connected open neighbourhood of the closed unit disk. Assume that $|f(z)|=|z+1|$ on the unit circle $|z|=1$, that $f(1)=2$, and that $f$ has simple zeros at $\pm i/2$ and no other zeros in the disk $|z|<1$. Show that these properties determine $f$ uniquely.
Can someone give me any hint?
My attempt was to use something like the Blaschke product decomposition bt $f$ doesn't necessarily map the disk to itself. I tried skimming through some books but I haven't found anything relevant to solve this question.