On Hayman Conjecture for Paired Complex Delay-Differential Polynomials

Abstract

We study Hayman conjecture for different paired complex polynomials under certain conditions. In 2021, the zeros distribution of $f^n(z)L(g)-a(z)$ and $g^n(z)L(f)-a(z)$ was studied by Gao and Liu for $n\ge 3$. In this paper, we work on the zeros distribution of $f^2(z)L(g)-a(z)$ and $g^2(z)L(f)-a(z)$, where $a(z)$ is a non-zero small function of both $f(z)$ and $g(z)$, and $L(h)$ takes the $k$th derivative $h^{(k)}(z)$ or shift $h(z+c)$ or difference $h(z+c)-h(z)$ or delay-difference $h^{(k)}(z+c)$, here $k\ge 1$ and $c$ is a non-zero constant. Moreover, we discuss Hayman conjecture for paired complex differential polynomials when $n=1.$