On the Zeros of Analytic Functions in the Disk with a Given Majorant near Its Boundary

Suppose that $\lambda$ is an arbitrary positive function from $C[0,1)$, such that $\lambda(r)\to\infty$ as $r\to 1-0$ and satisfying some growth regularity conditions, $A(\lambda)$ is the set of all holomorphic functions $f$ in the unit disk for which ${\ln}|f(z)|\le c\cdot\lambda(|z|)$, $|z|<1$. In this paper, we establish that there exists a function $f\in A(\lambda)$ with root set $\{z_k\}_{k=1}^{+\infty}$ such that the sequence $\{|z_k|\}_{k=1}^{+\infty}$ is the uniqueness set for the class $A(\lambda)$.