Solutions of Complex Differential Equation Having Pre-given Zeros in the Unit Disc

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.