On non-separated zero sequences of solutions of a linear differential equation

Abstract

Let $(z_k)$ be a sequence of distinct points in the unit disc $\mathbb{D}$ without limit points there. We are looking for a function $a(z)$ analytic in $\mathbb{D}$ and such that possesses a solution having zeros precisely at the points $z_k$, and the resulting function $a(z)$ has `minimal' growth. We focus on the case of non-separated sequences $(z_k)$ in terms of the pseudohyperbolic distance when the coefficient $a(z)$ is of zero order, but $\sup_{z\in \mathbb{D}} (1-|z|)^p |a(z)|=+\infty$ for any $p>0$. We established a new estimate for the maximum modulus of $a(z)$ in terms of the functions $n_z(t)=\sum_{|z_k-z|\le t} 1 $ and $N_z(r)=\int_0^r \frac{(n_z(t)-1)^+}{t}dt.$ The estimate is sharp in some sense. The main result relies on a new interpolation theorem.