Equivalence of summatory conditions along sequences for bounded holomorphic functions

Details Equivalence of summatory conditions along sequences for bounded holomorphic functions Equivalence of summatory conditions along sequences for bounded holomorphic functions

2003-06-26

arxiv.org/abs/math/0306376v2

Mathematics

Complex Variables

15 pages, LaTeX; some typos corrected. To appear in the issue of Complex Variables dedicated to the memory of Matts Essen

Scientific paper

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 $\sum_k \rho(1-|z_k|)|f(z_k)|$ 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.

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