Geodesic
Generalized pointwise H\"older type conditions of order less than two for an analytic function and its modulus
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 155-159

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The results of a recent paper by A. V. Vasin, S. V. Kislyakov, and the author on the relationship between the local boundary smoothness of an analytic function and local boundary smoothness of its modulus are extended to the case of generalized pointwise Hölder type conditions of order between one and two. 
@article{ZNSL_2017_456_a12,
     author = {A. N. Medvedev},
     title = {Generalized pointwise {H\"older} type conditions of order less than two for an analytic function and its modulus},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {155--159},
     publisher = {mathdoc},
     volume = {456},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_456_a12/}
}
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A. N. Medvedev. Generalized pointwise H\"older type conditions of order less than two for an analytic function and its modulus. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 155-159. http://geodesic.mathdoc.fr/item/ZNSL_2017_456_a12/