Nonuniqueness Sequences for Weighted Algebras of Holomorphic Functions in the Unit Circle

Abstract

Suppose that $\mathcal{P}$ is a system of continuous subharmonic functions in the unit disk $\mathbb{D}$ and $\mathbb{A}_\mathcal{P}$ is the class of holomorphic functions f in $\mathbb{D}$ such that log|f(z)| ≤ B _f p _f (z) + C _f , z ∈ $\mathbb{D}$, where B _f and C _f are constants and p _f ∈ $\mathcal{P}$. We obtain sufficient conditions for a given number sequence Λ = { λ_n} ⊂ $\mathbb{D}$ to be a subsequence of zeros of some nonzero holomorphic function from $\mathbb{A}_\mathcal{P}$, i.e., Λ is a nonuniqueness sequence for $\mathbb{A}_\mathcal{P}$.