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

Suppose that $\mathscr P$ is a system of continuous subharmonic functions in the unit disk $\mathbb D$ and $A_{\mathscr P}$ is the class of holomorphic functions $f$ in $\mathbb D$ such that $\log|f(z)|\le B_fp_f(z)+C_f$, $z\in\mathbb D$, where $B_f$ and $C_f$ are constants and $p_f\in\mathscr P$. We obtain sufficient conditions for a given number sequence $\Lambda=\{\lambda_n\}\subset\mathbb D$ to be a subsequence of zeros of some nonzero holomorphic function from $A_{\mathscr P}$, i.e., $\Lambda$ is a nonuniqueness sequence for $A_{\mathscr P}$.