Conformal representation

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[Complex analysis question]

I'm not sure if it is even possible to do conformal representation of function G to G*, since I need to transform one circle (line is circle with infinite radius) to two circles that interact.

If anyone can give me some opinion about this, it would be great since I don't have any clue how to do it and therefore I think it is even impossible.

Given functions;

$G = {y ≤ x}$ in $G^* = \{|z−i| > √2, |z + i| < √2\} $