Geodesic
Bounded holomorphic functions covering no concentric circles
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 29, Tome 429 (2014), pp. 34-43

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By the symmetrization method, the growth and distortion theorems for the functions mentioned in the title of the paper are proved. Precise estimates for the moduli of such functions and of their derivatives at inner and boundary points and an estimate for the Schwarzian derivative are given. 
@article{ZNSL_2014_429_a3,
     author = {V. N. Dubinin},
     title = {Bounded holomorphic  functions covering no concentric circles},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {34--43},
     publisher = {mathdoc},
     volume = {429},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_429_a3/}
}
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V. N. Dubinin. Bounded holomorphic  functions covering no concentric circles. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 29, Tome 429 (2014), pp. 34-43. http://geodesic.mathdoc.fr/item/ZNSL_2014_429_a3/