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 work analyzes the value distribution of differential-difference monomials generated by a transcendental entire function $\mathfrak{f}$ with hyper-order $\rho_2(\mathfrak{f})<1$. Using Nevanlinna theory and a suite of shift- and derivative-based lemmas for hyper-order $<1$, the authors derive two sharp inequalities: $T(r,\mathfrak{f})$ is controlled by reduced counting functions of $\alpha(z)\mathfrak{f}(z)^{q_0}(\mathfrak{f}(z+c))^{q_1} - a(z)$ and, under a derivative-augmented monomial, by $\alpha(z)\mathfrak{f}(z)^{q_0}(\mathfrak{f}'(z+c))^{q_1} - a(z)$, with explicit coefficients depending on $q_0$ and $q_1$. The results, proven under $q_0\ge2$, $q_1\ge1$, and $q_0>q_1+1$ respectively, extend Hayman-type and Xu–Yi–Zhang–type inequalities to differential-difference settings and rely on the asymptotic smallness of $\alpha(z)$ and $a(z)$. This advances the understanding of how nonlinear differential-difference operators constrain the value distribution of transcendental entire functions in the low hyper-order regime.

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 $α\mathfrak{f}(z)^{q_0}(\mathfrak{f}(z+c))^{q_1}$, where $c$ is a non-zero complex number and $q_0\geq2,$ $q_1\geq 1$ are non-negative integers, and $ α(z)$ $(\not\equiv 0,\infty)$ be a small function of $\mathfrak{f}$.

Theorems & Definitions (20)

• Definition 1.1
• Definition 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof