Geodesic
Conformal maps of a region that is geometrically close to a disk
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 50, Tome 512 (2022), pp. 116-147

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Let $D$ be a Jordan domain differing from the unit disk in a finite number of domains of small diameter, and let $f$ be a conformal mapping of $D$ onto the unit disk. Under some additional assumptions, the deviation of $f$ from the identity mapping is estimated in explicit terms. 
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     author = {M. S. Kuznetsova and N. A. Shirokov},
     title = {Conformal maps of a region that is geometrically close to a disk},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {116--147},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_512_a7/}
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M. S. Kuznetsova; N. A. Shirokov. Conformal maps of a region that is geometrically close to a disk. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 50, Tome 512 (2022), pp. 116-147. http://geodesic.mathdoc.fr/item/ZNSL_2022_512_a7/