Slowly growing solutions of ODEs revisited

TL;DR

This paper revisits the growth conditions of solutions to the differential equation $f''+Af=0$, exploring the relationship between the coefficient's behavior and the solutions' membership in function spaces like the Bloch space, BMOA, and VMOA.

Contribution

It demonstrates that existing coefficient conditions do not fully characterize when solutions are in the Bloch space and investigates the growth of the coefficient $A$ when all solutions are in $ ext{Bloch}$, $ ext{BMOA}$, or $ ext{VMOA}$.

Findings

Certain coefficient conditions fail to detect Bloch solutions.

The growth of $A$ is constrained when all solutions are in specific function spaces.

A revised perspective on slowly growing solutions in relation to function spaces.

Abstract

Solutions of the differential equation $f''+Af=0$ are considered assuming that $A$ is analytic in the unit disc $\mathbb{D}$ and satisfies \begin{equation} \label{eq:dag} \sup_{z\in\mathbb{D}} \, |A(z)| (1-|z|^2)^2 \log\frac{e}{1-|z|} < \infty. \tag{$\star$} \end{equation} By recent results in the literature, such restriction has been associated to coefficient conditions which place all solutions in the Bloch space $\mathcal{B}$. In this paper it is shown that any coefficient condition implying \eqref{eq:dag} fails to detect certain cases when Bloch solutions do appear. The converse problem is also addressed: What can be said about the growth of the coefficient $A$ if all solutions of $f''+Af=0$ belong to $\mathcal{B}$? An overall revised look into slowly growing solutions is presented, emphasizing function spaces $\mathcal{B}$, $\rm{BMOA}$ and $\rm{VMOA}$.