# Solutions of complex differential equation having zeros on pre-given sequences

## TL;DR

This paper investigates solutions to a complex differential equation with zeros on prescribed sequences, revealing new conditions under which solutions can have zeros not satisfying classical criteria and connecting zero-sequences with Carleson measures.

## Contribution

It demonstrates that solutions can have zeros outside the Blaschke condition and characterizes zero-sequences via Carleson measures and uniform separation, answering open questions.

## Key findings

- Existence of solutions with zeros not satisfying the Blaschke condition.
- Characterization of zero-sequences via Carleson measures and uniform separation.
- Improvement of classical results on non-normal functions and local univalence.

## Abstract

Behavior of solutions of $f''+Af=0$ is discussed under the assumption that $A$ is analytic in $\mathbb{D}$ and $\sup_{z\in\mathbb{D}}(1-|z|^2)^2|A(z)|<\infty$, where $\mathbb{D}$ is the unit disc of the complex plane. As a main result it is shown that such differential equation may admit a non-trivial solution whose zero-sequence does not satisfy the Blaschke condition. This gives an answer to an open question in the literature. It is also proved that $\Lambda\subset\mathbb{D}$ is the zero-sequence of a non-trivial solution of $f''+Af=0$ where $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure if and only if $\Lambda$ is uniformly separated. As an application an old result, according to which there exists a non-normal function which is uniformly locally univalent, is improved.