Converse growth estimates for ODEs with slowly growing solutions

Abstract

Let $f_1,f_2$ be linearly independent solutions of $f''+Af=0$, where the coefficient $A$ is an analytic function in the open unit disc $\mathbb{D}$ of $\mathbb{C}$. It is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function $u=-\log\, (f_1/f_2)^{\#}$. For example, the case when $\sup_{z\in\mathbb{D}} |A(z)|(1-|z|^2)^2 < \infty$ and $f_1/f_2$ is normal, is characterized by the condition $\sup_{z\in\mathbb{D}} |\nabla u(z)|(1-|z|^2) < \infty$. Different types of Blaschke-oscillatory equations are also described in terms of harmonic majorants of $u$. Even if $f_1,f_2$ are bounded linearly independent solutions of $f''+Af=0$, it is possible that $\sup_{z\in\mathbb{D}} |A(z)|(1-|z|^2)^2 = \infty$ or $f_1/f_2$ is non-normal. These results relate to sharpness discussion of recent results in the literature, and are succeeded by a detailed analysis of differential equations with bounded solutions. Analogues results for the Nevanlinna class are also considered, by taking advantage of Nevanlinna interpolating sequences. It is shown that, instead of considering solutions with prescribed zeros, it is possible to construct a bounded solution of $f''+Af=0$ in such a way that it solves an interpolation problem natural to bounded analytic functions, while $|A(z)|^2(1-|z|^2)^3\, dm(z)$ remains to be a Carleson measure.