Growth of Solutions of Complex Differential Equations with Entire Coefficients having a Multiply-Connected Fatou Component

TL;DR

This paper investigates the growth properties of solutions to complex differential equations with entire coefficients, demonstrating that solutions have infinite order under specific conditions related to the coefficients' Fatou components.

Contribution

It establishes that solutions of certain linear differential equations with entire coefficients possessing multiply-connected Fatou components are of infinite order, extending previous results to higher-order equations.

Findings

Solutions have infinite order when $A(z)$ meets certain restrictions and $B(z)$ has a multiply-connected Fatou component.

Results apply to higher-order linear differential equations.

Provides conditions linking Fatou components to solution growth.

Abstract

In this study, we show that all non-trivial solutions of $f"+A(z)f'+B(z)f=0$ have infinite order, provided that the entire coefficient $A(z)$ has certain restrictions and $B(z)$ has multiply-connected Fatou component. We also extend these results to higher order linear differential equations.