On the boundary behaviour of certain classes of functions (Q2764619)

This is a survey of results dealing with the boundary behavior of classes of holomorphic and harmonic functions in the unit ball of the complex plane. There is also some brief discussion of the situation in the unit ball in the complex \(n\)-space. Although the survey covers a wide variety of results, many of the results mentioned focus on the set \({\mathcal A}(f)\) of points on the boundary \(T\) of the unit ball at which asymptotic values occur. A common thread of many of the results deals with circumstances under which the set \({\mathcal A}(f)\) is non-negligeable on an arc \(I\subset T\), that is, \({\mathcal A}(f)\cap I\) has either positive measure, Hausdorff dimension, or positive capacity, as appropriate to the hypotheses. Results mentioned range from classical results (for example, due to Plessner, Bagemihl, and G. R. MacLane) to relatively recent results (for example, due to \textit{J. J. Carmona} and \textit{Ch. Pommerenke} [J. Lond. Math. Soc., II. Ser. 56, No. 1, 16-36 (1997; Zbl 0892.30004)], and \textit{J. J. Betancor}, \textit{J. C. Farina}, and \textit{J. G. Llorente} [Potential Analysis (to appear)]). An extensive list of references is given.