We provide results of uniqueness for holomorphic functions in the Nevanlinnaclass bridging those previously obtained by Hayman and Lyubarskii-Seip. Namely,we propose certain classes of hyperbolically separated sequences in the disk,in terms of the rate of non-tangential accumulation to the boundary (theendpoints of this spectrum of classes being respectively the sequences with anon-tangential cluster set of positive measure, and the sequences violating theBlaschke condition); and for each of those classes, we give a criticalcondition of radial decrease on the modulus which will force a Nevanlinna classfunction to vanish identically.
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