Linear differential equations with slowly growing solutions

TL;DR

This paper investigates conditions on coefficients for linear differential equations in the unit disc, analyzing zero separation, solution growth in various function spaces, and extending Hardy space formulas to higher derivatives.

Contribution

It introduces new coefficient conditions for zero separation and solution growth, and extends Hardy-Stein-Spencer formulas to higher derivatives in the context of differential equations.

Findings

Zeros of solutions are separated under certain coefficient conditions

Solutions exhibit slow growth in $H^ Infty$, BMOA, and Bloch spaces

A Hardy-Stein-Spencer type formula for higher derivatives is established

Abstract

This research concerns coefficient conditions for linear differential equations in the unit disc of the complex plane. In the higher order case the separation of zeros (of maximal multiplicity) of solutions is considered, while in the second order case slowly growing solutions in $H^\infty$, $\rm{BMOA}$ and the Bloch space are discussed. A counterpart of the Hardy-Stein-Spencer formula for higher derivatives is proved, and then applied to study solutions in the Hardy spaces.