On oscillation of solutions of linear differential equations

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.