On oscillation of solutions of linear differential equations

TL;DR

This paper investigates the zeros of solutions to linear differential equations in the complex plane, relating their distribution to the boundary behavior of the coefficient and exploring conditions for finite oscillation.

Contribution

It establishes a relationship between zero accumulation points and boundary behavior of the coefficient, and shows zero distribution restrictions under a specific coefficient bound.

Findings

Zeros' accumulation points relate to boundary behavior of A

Bound on A restricts zero distribution of solutions

Open problem: does the bound imply finite oscillation?

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.