On value distribution of certain delay-differential polynomials

TL;DR

This paper investigates the zero distribution of certain delay-differential polynomials of entire functions of finite order, providing improved results that extend previous work in the field.

Contribution

It offers new insights into the zero distribution of delay-differential polynomials, extending and refining earlier results by Chen and Laine.

Findings

Improved bounds on zero distribution for delay-differential polynomials.

Extended results for cases n≥3 and n=2.

Enhanced understanding of the value distribution of such functions.

Abstract

Given an entire function $f$ of finite order $\rho$, let $L(z,f)=\sum_{j=0}^{m}b_{j}(z)f^{(k_{j})}(z+c_{j})$ be a linear delay-differential polynomial of $f$ with small coefficients in the sense of $O(r^{\lambda+\varepsilon})+S(r,f)$, $\lambda<\rho$. Provided $\alpha$, $\beta$ be similar small functions, we consider the zero distribution of $L(z,f)-\alpha f^{n}-\beta$ for $n\geq 3$ and $n=2$, respectively. Our results are improvements and complements of Chen(Abstract Appl. Anal., 2011, 2011: ID239853, 1--9), and Laine (J. Math. Anal. Appl. 2019, 469(2): 808--826.), etc.