A boundary uniqueness theorem for regular functions with bounded integral of Dirichlet type

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.