On the Goldberg-Ostrovskii Problem for Linear Differential Equations with Exponential Polynomial Coefficients

TL;DR

This paper proves that finite-order solutions of linear differential equations with exponential polynomial coefficients inherit regular growth properties, confirming a conjecture for this specific class of coefficients.

Contribution

It establishes that exponential polynomial coefficients ensure the inheritance of regular growth in solutions, affirming a conjecture and advancing understanding of function class structures.

Findings

Regularity transfer holds for exponential polynomial coefficients.

Counterexamples exist for general coefficients, but not for exponential polynomials.

The results highlight the closed nature of exponential polynomials in growth transfer.

Abstract

The Goldberg-Ostrovskii problem asks whether finite-order solutions of a linear differential equation inherit the property of completely regular growth (c.r.g.) from its coefficients. While Bergweiler's counterexample demonstrated that the answer is negative in general, this paper proves that when the coefficients are restricted to the classical and rich subclass of exponential polynomials, the regularity transmission does hold. Thereby we affirm the conjecture posed by Heittokangas, Ishizaki, Tohge and Wen. Our results reveal the closed nature of exponential polynomials in the context of regularity transfer from the perspective of equation dynamics, and provide a new perspective for the study of the structure of related function classes.