Equivalence of summatory conditions along sequences for bounded holomorphic functions

A sequence of points $z_k$ in the unit disk is said to be thin for a given decrease function $\rho$, if there is a nontrivial bounded holomorphic function such that the infinite series ${\sum }_k\rho (1-\mathrm{\mid }{z}_k\mathrm{\mid })\mathrm{\mid }f(z_k)\mathrm{\mid }$ converges. All sequences will be assumed hyperbolically separated. We give necessary and sufficient conditions for the problem of thinness of a sequence to be non-trivial (one way or the other), and for two different decrease functions to give rise to the same thin sequences. Along the way, some concrete conditions (necessary or sufficient) for a sequence to be thin are obtained.