Geodesic
Drop of the smoothness of an outer function compared to the smoothness of its modulus, under restrictions on the size of boundary values
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 43, Tome 434 (2015), pp. 101-115

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Let $F$ be an outer function on the unit disk. It is well known that its smootheness properties may be two times worse that those of the modulus of its boundary values, but under some restrictions on $\log|F|$ this gap becomes smaller. It is shown that the smoothness decay admits a convenient description in terms of a rearrangement invariant Banach function space containing $\log|F|$. All results are of pointwise nature. 
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     author = {A. N. Medvedev},
     title = {Drop of the smoothness of an outer function compared to the smoothness of its modulus, under restrictions on the size of boundary values},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {101--115},
     publisher = {mathdoc},
     volume = {434},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_434_a8/}
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A. N. Medvedev. Drop of the smoothness of an outer function compared to the smoothness of its modulus, under restrictions on the size of boundary values. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 43, Tome 434 (2015), pp. 101-115. http://geodesic.mathdoc.fr/item/ZNSL_2015_434_a8/