Finite Logarithmic Order Meromorphic Solutions of Complex Linear Delay-Differential Equations

TL;DR

This paper investigates the growth of meromorphic solutions with finite logarithmic order for complex linear delay-differential equations, extending recent results to include the logarithmic lower order.

Contribution

It extends existing results on the growth of solutions to delay-differential equations by incorporating the concept of logarithmic lower order.

Findings

Solutions have finite logarithmic order.

Extended growth estimates to include logarithmic lower order.

Generalized previous results by Chen, Zheng, Bellaama, and Bela"{\

Abstract

In this article, we study the growth of meromorphic solutions of linear delay-differential equation of the form \begin{equation*} \sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}(z)f^{(j)}(z+c_{i})=F(z), \end{equation*}% where $A_{ij}(z)$ $(i=0,1,\ldots ,n,j=0,1,\ldots ,m,n,m\in \mathbb{N})$ and $% F(z)$ are meromorphic of finite logarithmic order, $c_{i}(i=0,\ldots ,n)$ are distinct non-zero complex constants. We extend those results obtained recently by Chen and Zheng, Bellaama and Bela\"{\i}di to the logarithmic lower order.