Determine an analytic function over the disk 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.

According to the given details we can write $f(z)=(z+1)g(z)$ for some $g$ : analytic function in a neighborhood of closed unit disk with $g(\pm i/2)=0$ and $g(1)=1$ and $|g(z)|=1$ on the unit circle. Therefore as you expected $g$ is a finite Blaschke product that vanishes at $\pm i/2.$ Thus $g(z)=c\left(\dfrac{4z^2+1}{z^2+4}\right)$ for some uni-modular constant $c.$ Also evaluating this function at $1,$ we can obtain $c=1.$ Hence $$f(z)=(z+1)\left(\dfrac{4z^2+1}{z^2+4}\right).$$
