Biholomorphic map from a disk to its quarter

Question

Could you tell me how to find a biholomorphic map from a unit disk $D$ to $\{ |z|<1 \ : \ \Re z >0, \ \Im z >0 \}$?

I know that mapping of the form : $\frac{az+b}{cz+d}$ won't work.

Also, we can map the quarter of the disk to upper half of the disk by $f(z) = z^2$.

Then, using the map $g(z)= \frac{z-1}{z+1}$ we map $Q:= \{ 0< \arg z < \frac{\pi}{2}\}$ to $D_+$ the upper half of the unit disk.

So $g^{-1}$ maps $D_+$ to $Q$

And $\varphi (z) = \frac{iz^2+1}{iz^2-1}: \ Q \rightarrow D$

Will this mapping work?

$f^{-1} \circ g \circ \varphi ^{-1} (z)$

Answer 1

There is a mistake (probably a typo) in the last line: it should be $ f^{-1}\circ g\circ\varphi^{-1}$. The rest is correct.