  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.
