On the boundary behaviour of functions in the Djrbashyan classes \(U_{\alpha}\) and \(A_{\alpha}\)

Abstract

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 \(z \to e^{i\theta}\).