A Note on the value distribution of some differential-difference monomials generated by a transcendental entire function of hyper-order less than one

TL;DR

This paper investigates the value distribution of certain differential-difference monomials generated by transcendental entire functions with hyper-order less than one, extending understanding of their value distribution properties.

Contribution

It provides new results on the value distribution of differential-difference monomials involving transcendental entire functions of hyper-order less than one.

Findings

Established bounds on the value distribution of the monomials.

Extended previous results to functions with hyper-order less than one.

Analyzed the role of small functions in the value distribution.

Abstract

Let $\mathfrak{f}$ be a transcendental entire function with hyper-order less than one. The aim of this note is to study the value distribution of the differential-difference monomials $\alpha \mathfrak{f}(z)^{q_0}(\mathfrak{f}(z+c))^{q_1}$, where $c$ is a non-zero complex number and $q_0\geq2,$ $q_1\geq 1$ are non-negative integers, and $ \alpha(z)$ $(\not\equiv 0,\infty)$ be a small function of $\mathfrak{f}$.