Geodesic
A~boundary uniqueness theorem for regular functions with bounded integral of Dirichlet type
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 180-190

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Closed sets of uniqueness on $\partial U$ for a class of analytical functions in the unit circle $U$ for which $$ \iint_U|f'(z)|^2h(z)\,d\sigma+\infty. $$ are considered in the paper. The main result of the paper makes it possible to construct rather small closed sets of uniqueness for the class of functions involved. 
@article{ZNSL_1983_126_a19,
     author = {S. P. Preobrazenskii},
     title = {A~boundary uniqueness theorem for regular functions with bounded integral of {Dirichlet} type},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {180--190},
     publisher = {mathdoc},
     volume = {126},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_126_a19/}
}
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S. P. Preobrazenskii. A~boundary uniqueness theorem for regular functions with bounded integral of Dirichlet type. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 180-190. http://geodesic.mathdoc.fr/item/ZNSL_1983_126_a19/