On the boundary behaviour of functions in the Djrbashyan classes $U_\alpha$ and $A_\alpha$

Nevanlinna factorization theorem was essentially extended in a series of papers by M. M. Djrbashyan for classes $A_\alpha$ and $U_\alpha$ introduced by him, see [2], [3]. In this paper we pay particular attention to non vanishing functions $f\in A_\alpha(-1<\alpha<0)$ and show that for any $\theta$ except at most a set of zero $(1+\alpha)$-capacity we have $|\ln|f(z)||=o((1-|z|)^{1+\alpha})$ as $z\to e^{i\theta}$.