How do I know a function is a fractional linear transformation?

Question

I'm asked to find a nonconstant function from an open unit dist to an open unit disk which analytic on all of $D= \{z \in \mathbb{C}: |z|<1 \}$ with $g(1/2)=0$ and $g(1/3)=0$.

Here's what I put: $$\frac{2z-1}{2-z}*\frac{3w-1}{3-w}$$

Now here's the question I'm asked to answer: Can this function be a fractional linear transformation? Why or why not?

My guess is: yes? I mean that's a product of two fractional transformations which implies a Mobius transformation? I don't know the answer to this really if someone can please help clarify, I'd appreciate it. Thank you.

Answer 1

The product(not composition) of two fractional transformations is not a fractional transformation. There is no such fractional linear transformation mapping $\frac{1}{2}$ and $\frac{1}{3}$ to $0$, for a linear fraction has at most 1 root.