Geodesic
Estimates of parameters of conformal mappings related to a~periodic Jacobi matrix
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 32, Tome 315 (2004), pp. 102-120

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We study the conformal mappings of the band without some vertical symmetric slits into the plane without horizontal slits (which lie on the real axis). Sharp two-sided estimates involving the lengths of slits and other parameters are obtained. In terms of the spectral theory of Jacobi matrices, we obtain sharp estimates involving isospectral parameters and the norm of potentials. 
@article{ZNSL_2004_315_a7,
     author = {A. Kutsenko},
     title = {Estimates of parameters of conformal mappings related to a~periodic {Jacobi} matrix},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {102--120},
     publisher = {mathdoc},
     volume = {315},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_315_a7/}
}
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A. Kutsenko. Estimates of parameters of conformal mappings related to a~periodic Jacobi matrix. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 32, Tome 315 (2004), pp. 102-120. http://geodesic.mathdoc.fr/item/ZNSL_2004_315_a7/