As an elementary result in Complex Analysis, one can use the Argument Principal to show that the Schwarz-Christoffel transform is injective on the interior of the polygon to which it maps. Could this one-to-one correspondence be used to show that such an inverse mapping exists for at least a subset of the upper half-plane? In my search for an answer, I found page 24 of the following [paper] that mentions the existence of a trigonometric inverse for a mapping to a bar-shaped region. Does a generalization of this result hold? More specifically, has existing literature produced an inverse formula for any polygonal geometry? 

I should also hope to refer anyone curious about the existence and Holder continuity of an inverse to the following [post].

[post]: https://mathoverflow.net/questions/352430/inverse-of-the-schwartz-christoffel-map-and-the-continuity?rq=1

Thank you all. 

[![Inverse Schwarz-Christoffel Mapping][1]][1]

[paper]: https://fse.studenttheses.ub.rug.nl/8761/1/Willem_Hendriks_WB_2009.pdf


  [1]: https://i.sstatic.net/eZ6o8.png