0
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ This is the same question of part C) from this question $\endgroup$ – user486983 Dec 22 '19 at 0:54
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.