What is the density of distribution which is obtained by acting with a Mobius transformation on the unit disc with uniform distribuition?

Question

So, I have given a Mobius transformation that preserves unit disc. In unit disc I have uniform distribution (distribution that has constant density). I have to act with this Mobius transformation on this uniform distribution. What will I get?What density it will have?

Can anybody please help me with this? Or at least tell me which literature available on internet I should look that would give me the best answer on this... Thank you!

Answer 1

A map $z\to f(z)$ expands the area by a factor $|f'(z)|^2$. Thus the density at $f(z)$ is proportional to $\frac1{|f'(z)|^2}$. Equivalently, the density at $w$ is proportional to $|{f^{-1}}'(w)|^2$.