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

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 $q_0\ge 2,$ $q_1\ge 1$ are non-negative integers, and $\alpha (z)$ $(\not\equiv 0,\infty )$ be a small function of $\mathfrak{f}$.