Entire solutions of delay differential equations of Malmquist type

TL;DR

This paper characterizes entire solutions of a class of delay differential equations of Malmquist type, identifying conditions for solutions with low hyper-order and analyzing their growth and value distribution.

Contribution

It provides a complete characterization of reduced forms of these equations admitting transcendental entire solutions with hyper-order less than one, extending previous results.

Findings

Identifies all reduced forms with entire solutions of hyper-order less than one.

Analyzes growth order and value distribution of solutions.

Removes previous assumptions about roots of the denominator in rational functions.

Abstract

In this paper, we investigate the delay differential equations of Malmquist type of form \begin{equation*} w(z+1)-w(z-1)+a(z)\frac{w'(z)}{w(z)}=R(z, w(z)),~~~~~~~~~~~~~~(*) \end{equation*} where $R(z, w(z))$ is an irreducible rational function in $w(z)$ with rational coefficients and $a(z)$ is a rational function. We characterize all reduced forms when the equation $(*)$ admits a transcendental entire solutions with hyper-order less than one. When we compare with the results obtained by Halburd and Korhonen[Proc.Amer.Math.Soc., forcoming],we obtain the reduced forms without the assumptions that the denominator of rational function $R(z,w(z))$ has roots that are nonzero rational functions in $z$. The growth order and value distribution of transcendental entire solutions for the reduced forms are also investigated.