M. Langer and H. Woracek,
Stability of the derivative of a canonical product,
Complex Anal. Oper. Theory 8 (2014), 1183–1224

Abstract:
With each sequence α=(αn)n∈ℕ of pairwise distinct and non-zero points which are such that the canonical product

Pα(z) := limr→∞n|≤r (1 – zn)
converges, the sequence
α' := (Pα'(αn))n∈ℕ
is associated. We give conditions on the difference β – α of two sequences which ensure that β' and α' are comparable in the sense that
c,C > 0:   c|α'n| ≤ |β'n| ≤ C|α'n|,   n ∈ ℕ.
The values α'n play an important role in various contexts. As a selection of applications we present: an inverse spectral problem, a class of entire functions and a continuation problem.

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