On value distribution of certain delay-differential polynomials

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 $O(r^{\lambda +\epsilon })+S(r,f)$, $\lambda <\rho$. Provided $\alpha$, $\beta$ be similar small functions, we consider the zero distribution of $L(z,f)-\alpha f^n-\beta$ for $n\ge 3$ 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.