On non-normal solutions of linear differential equations

TL;DR

This paper investigates the oscillation behavior of solutions to a specific class of linear differential equations with analytic coefficients in the unit disc, focusing on solutions that are non-normal yet have prescribed zeros.

Contribution

It introduces conditions under which non-normal solutions with prescribed zeros exist for differential equations with bounded coefficient growth.

Findings

Non-normal solutions can have prescribed uniformly separated zeros.

Normality arguments help understand oscillation of solutions.

Existence of solutions with specific zero distributions is established.

Abstract

Normality arguments are applied to study the oscillation of solutions of $f''+Af=0$, where the coefficient $A$ is analytic in the unit disc $\mathbb{D}$ and $\sup_{z\in\mathbb{D}} (1-|z|^2)^2|A(z)| < \infty$. It is shown that such differential equation may admit a non-normal solution having prescribed uniformly separated zeros.