If $f$ is analytic on a domain $D$ and it's mapping the region onto a circle, then what can be said about $f$?
I tried to write $f(z)$ as $(r\cos\theta+a)+(r\sin\theta+b)i$, and by using the Cauchy Riemann equations I get that it must be a unit circle if the function is analytic since cosθ=rcosθ and sinθ=-(-rsinθ)
I think by using the open mapping theorem, one can easily state that $f$ is constant because a circle is not an open set.
It that right? But what can I do if I don't the theorem? Need some hint.