On Hayman Conjecture for Paired Complex Delay-Differential Polynomials

TL;DR

This paper investigates the zeros distribution of paired complex polynomials involving derivatives, shifts, and differences, extending previous results to new cases and exploring the Hayman conjecture for these polynomial pairs.

Contribution

It advances the understanding of the Hayman conjecture by analyzing zeros of specific paired complex polynomials with derivatives, shifts, and differences, including new cases for n=2 and n=1.

Findings

Zeros distribution results for $f^{2}(z)L(g)-a(z)$ and $g^{2}(z)L(f)-a(z)$

Extension of Hayman conjecture to paired differential polynomials with $n=1$

Conditions under which zeros are distributed in these polynomial pairs

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 $n\geq 3$. 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 $k\geq 1$ and $c$ is a non-zero constant. Moreover, we discuss Hayman conjecture for paired complex differential polynomials when $n=1.$