I am trying to find a complex mapping from the left-hand side of the plane to the unit disk, such that $$\lvert z_1\rvert\gt\lvert z_2\rvert\iff \lvert f(z_1)\rvert\gt\lvert f(z_2)\rvert$$ I wasn't sure what to call this property, so the title is a bit weird. According to this answer
there is a holomorphic bijection from the open unit disk onto a region $U$ if and only if $U$ is simply connected and the complement of $U$ (in the Riemann sphere) has at least two points.
The Möbius transformation $w=\frac{z+1}{z-1}$ maps the left side into the unit disk, but it doesn't satisfy that property. I tried some other forms of the bilinear transformation without any luck, and I suspect that such transformation may not exist. Is my suspicion correct?
In the title you didn't say 'onto' but in the question you have put this extra condition. Since $f$ is onto there exists $z_2$ such that $w_2=f(z_2)=0$. But then $|z_1|<|z_2|$ implies $|w_1|<0$, a contradiction.