On the angular derivative of analytic functions (Q1821229)

The paper deals with the relations between the existence of tangents (defined in a completely geometric way) and the existence of angular derivatives. More precisely, a function f analytic in the unit disk and its global cluster set B are considered. The main result is the following: if B has tangents at a ''large'' subset of the boundary of the image domain, then f' has angular limits almost everywhere. In particular, the assumptions of the theorem are satisfied whenever the linear measure of B is finite and positive. For Hadamard gap series this yields some new characterizations of f' belonging to the Hardy space \(H^ 1\).