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

Abstract

Behavior of solutions of $f^{\prime \prime }+Af=0$ 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 $f^{\prime \prime }+Af=0$ 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.

11 pages

References in corpus (1)

• Uniform separation through intermediate points

Cited by in corpus (1)

• Converse growth estimates for ODEs with slowly growing solutions