Abstract:
The class of functions with finite Dirichlet-type integral is defined as the set of holomorphic functions $f$ in the unit disk satisfying the following condition:
$$
\int_{0}^{2\pi}\int_{0}^{1} (1-r)^{\alpha}|f'(re^{i\theta})|^{p} r\,dr\,d\theta,\qquad \alpha>-1,\quad 0<p<+\infty.
$$
These classes are usually denoted by $D_{\alpha}^p$. In this paper, we prove the converse of Rudin's theorem and thus provide a necessary and sufficient condition for a Blaschke product to belong to the class $D_{0}^{1}$.
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