Geodesic
Keldysh--Lavrentiev counterexample and means of powers of conformal mapping. 1.~Construction of a~mapping
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 252-256

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We construct a region with infinitely many cusps on its boundary such that it is possible to estimate the derivative of a conformal mapping of the unit disk onto this region from below. We use here some ideas taken from a construction of a famous Keldysh–Lavrentiev example. 
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     author = {H. Hedenmalm and N. A. Shirokov},
     title = {Keldysh--Lavrentiev counterexample and means of powers of conformal mapping. {1.~Construction} of a~mapping},
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H. Hedenmalm; N. A. Shirokov. Keldysh--Lavrentiev counterexample and means of powers of conformal mapping. 1.~Construction of a~mapping. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 252-256. http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a13/