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 there. We are looking for function $a(z)$ analytic and such that possesses solution having zeros precisely at $z_k$ , resulting has ‘minimal’ growth. focus on case non-separated sequences terms pseudohyperbolic distance when coefficient is zero order, but $\sup _{z\in {\mathbb D}}(1-|z|)^p|a(z)| = + \infty$ any $p > 0$ . established new estimate maximum modulus functions $n_z(t)=\sum \nolimits _{|z_k-z|\le t} 1$ $N_z(r) \int_0^r {{(n_z(t)-1)}^ } /t{\rm d}t.$ The sharp some sense. main result relies interpolation theorem.