Stability of sequences of zeros for classes of holomorphic functions of moderate growth in the unit disk

Let $\Lambda=(\lambda_k)$ and $\Gamma=(\gamma_k)$ be two sequences of points in the unit disk $\mathbb D:=\{z\in\mathbb C\colon|z|<1\}$ of the complex plane $\mathbb C$, and $H$ be a weight space of holomorphic functions on $\mathbb D$. Suppose that $\Lambda$ is the zero subsequence of some nonzero function from $H$. We give conditions of closeness of the sequence $\Gamma$ to the sequence $\Lambda$, under which the sequence $\Gamma$ is the zero sequence for some holomorphic function from space $\hat H\supset H$. The space $\hat H$ can be a little larger than $H$.