Mean growth and geometric zero distribution of solutions of linear differential equations

TL;DR

This paper investigates conditions on the coefficient of a linear differential equation in the unit disc that ensure solutions have controlled growth and zero distribution, using Carleson measures.

Contribution

It introduces new Carleson measure conditions on the coefficient that guarantee solutions are in Hardy spaces or have uniformly separated zeros.

Findings

Solutions belong to Hardy spaces under certain coefficient conditions

Zero sequences of solutions are uniformly separated when conditions are met

Provides criteria linking coefficient properties to solution behavior

Abstract

The aim of this paper is to consider certain conditions on the coefficient $A$ of the differential equation $f"+Af=0$ in the unit disc, which place all normal solutions $f$ to the union of Hardy spaces or result in the zero-sequence of each non-trivial solution to be uniformly separated. The conditions on the coefficient are given in terms of Carleson measures.