Define \[ G \definedas \theset{z\in \CC\suchthat \Re(z) > 0, \, \abs{z-1} > 1} .\]
Find all of the injective conformal maps \(G\to \DD\). These may be expressed as compositions of maps, but explain why this list is complete.
Solution
Interpreting a conformal map \(G\to\mathbb D\) here as a biholomorphism onto \(\mathbb D\), first construct one such map. Put \[ T(z)=\frac1z. \] Since \[ \operatorname{Re}\frac1z=\frac{\operatorname{Re}z}{|z|^2}, \] the condition \(\operatorname{Re}z>0\) becomes \(\operatorname{Re}T(z)>0\). Moreover \[ |z-1|>1 \iff |z|^2>2\operatorname{Re}z \iff \operatorname{Re}\frac1z<\frac12. \] Thus \(T\) maps \(G\) biholomorphically onto the vertical strip \[ S=\{w:0<\operatorname{Re}w<1/2\}. \] Next \[ E(w)=e^{2\pi i w} \] maps \(S\) biholomorphically onto the upper half-plane, and the Cayley map \[ C(\zeta)=\frac{\zeta-i}{\zeta+i} \] maps the upper half-plane biholomorphically onto \(\mathbb D\). Hence \[ \Phi(z)=C(E(T(z))) =\frac{e^{2\pi i/z}-i}{e^{2\pi i/z}+i} \] is one biholomorphism \(G\to\mathbb D\).
Every disk automorphism has the form \[ A_{a,\theta}(w)=e^{i\theta}\frac{w-a}{1-\bar a w}, \qquad a\in\mathbb D,\quad \theta\in\mathbb R. \] Therefore all conformal bijections \(G\to\mathbb D\) are \[ \boxed{A_{a,\theta}\circ\Phi}. \] Indeed, if \(F:G\to\mathbb D\) is any other biholomorphism, then \(F\circ\Phi^{-1}\) is an automorphism of \(\mathbb D\), so it is one of the \(A_{a,\theta}\) above. This also proves completeness of the list.
If ``injective conformal map into \(\mathbb D\)’’ is read without the usual onto requirement, the displayed family is not exhaustive (for example \(r\Phi\), \(0<r<1\), is injective but not onto); the classification above is the standard conformal-equivalence interpretation of the question.