Given the open vertical strip $G=\{x+iy~|~0<x<1,~-\infty<y<\infty\}$, what is the explicit conformal injective map characterizing $w=f(z):G\to\mathbb{D}$?
It is noted that if there exists a mapping $w=M(z):G\to\mathbb{H}$, where $\mathbb{H}=G=\{x+iy~|y>0, (x,y)\in\mathbb{R}\}$. Therefore, the composition of $w=M(z)$ with the Cayley transform given by $\kappa(z):z\to \frac{z+i}{z-i}$, namely $\kappa(w(z)):w\to \frac{w+i}{w-i}$ is such a conformal mapping. However, what is the particular $w=M(z):G\to\mathbb{H}$?