Geodesic
Comparison of boundary smoothness for an analytic function and for its modulus in the case of the upper half-plane
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 75-89

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The results of a recent paper by A. V. Vasin, S. V. Kislyakov, and the author are extended to the case of outer functions in the upper half-plane. As in the case of the disk, it can only be guaranteed that the smoothness of an outer function is at least one half as high as that of it modulus, but the quantitative manifestation of this effect is different – in particular, it depends on the position of the point at which smoothness is measured. 
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     author = {A. N. Medvedev},
     title = {Comparison of boundary smoothness for an analytic function and for its modulus in the case of the upper half-plane},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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A. N. Medvedev. Comparison of boundary smoothness for an analytic function and for its modulus in the case of the upper half-plane. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 75-89. http://geodesic.mathdoc.fr/item/ZNSL_2016_447_a5/