How can you tell if a conformal mappimg between regions is unique? I have a conformal mapping from {z : |z|<2, |Arg(z)|< pi/6} to {z : Re(z)>0, Im(z)<0} as
f(z) = -(iz^3 - 8)/(iz^3 + 8) but have no idea how to find out if the conformal mappimg is unique? Any help is much appreciated.
Such a conformal mapping will not be unique in general.
For example, suppose $A$ and $B$ are two nonempty, proper, simply connected open subsets of $\Bbb C$, and suppose that $f$ is a conformal mapping from $A$ to $B$.
By the Riemann mapping theorem, both $A$ and $B$ are conformally equivalent to the open unit disk $D$. In particular, there is a conformal mapping $g\colon A\to D$.
The open unit disk also has a whole bunch of conformal automorphisms, given by Möbius transformations, let $m$ be one such that is not the identity map. (A simple example, if we want to be concrete, is $m(z)=-z$.)
It follows that $f$ and $f\circ g^{-1}\circ m\circ g$ are distinct conformal mappings from $A$ to $B$.
(Indeed, for such domains, the set of conformal mappings from $A$ to $B$ actually forms an infinite group that is isomorphic to the group of conformal automorphisms of $D$.)