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arXiv · 1506.00009

On oscillation of solutions of linear differential equations

Janne Gröhn·Jouni Rättyä
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Abstract

An interrelationship is found between the accumulation points of zeros of non-trivial solutions of $f"+Af=0$ and the boundary behavior of the analytic coefficient $A$ in the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$. It is also shown that the geometric distribution of zeros of any non-trivial solution of $f"+Af=0$ is severely restricted if $$\label{eq:cs_a}\tag{$\star$} |A(z)| (1-|z|^2)^2 \leq 1 + C (1-|z|), \quad z\in\mathbb{D}, $$ for any constant $0<C<\infty$. These considerations are related to the open problem whether \eqref{eq:cs_a} implies finite oscillation for all non-trivial solutions.

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Janne Gröhn, Jouni Rättyä. 2015-05-29. On oscillation of solutions of linear differential equations. https://doi.org/10.1007/s12220-016-9701-3

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