The characterization of conformal maps of the upper halfplane on a “comb” type domain

The domain $\{z\in\mathbf C: -\infty\leq a<\operatorname{Re}z<b\leq+\infty,\operatorname{Im}z>0\}\setminus\{\cup[x_k,x_k+iy_k]\}$ is called a “comb” type domain. For each closed set $E$ on the real axis there exists the unique conformal map of the upper halfplane onto a certain “comb” type domain of mapping the set $E$ on the interval $(a,b)$. If $a=-\infty$ and $b=+\infty$, then the set $E$ is referred to the type $(A)$. If either $a=-\infty$, $b<+\infty$, or $a>-\infty$, $b=+\infty$, then $E$ is referred to the type $(B)$. If both $a$ and $b$ are finite, then $E$ is referred to the type $(C)$. Conditions for a set $E$ to be referred to the type $(A)$, $(B)$ or $(C)$ are given.