Geodesic
The characterization of conformal maps of the upper halfplane on a “comb” type domain
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (1996) no. 3, pp. 290-307 
A. V. Kesarev. The characterization of conformal maps of the upper halfplane on a “comb” type domain. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (1996) no. 3, pp. 290-307. http://geodesic.mathdoc.fr/item/JMAG_1996_3_3_a5/
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     author = {A. V. Kesarev},
     title = {The characterization of conformal maps of the upper halfplane on a {\textquotedblleft}comb{\textquotedblright} type domain},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
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     year = {1996},
     volume = {3},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/JMAG_1996_3_3_a5/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The domain $\{z\in\mathbf C: -\infty\leq a<\operatorname{Re}z0\}\setminus\{\cup[x_k,x_k+iy_k]\}$ 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.