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Solutions of complex differential equation having zeros on pre-given sequences (1702.01585v1)

Published 6 Feb 2017 in math.CV and math.CA

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.

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