Entire Functions and Their High Order Difference Operators

Abstract

In this paper, we prove that for a transcendental entire function \(f\) of finite order such that \(\lambda(f-a)<\rho(f)\), where \(a\) is an entire function and satisfies \(\rho(a)<\rho(f)\), \(n\in\mathbb{N}\), if \(\Delta_{c}^{n}f\) and \(f\) share the entire function \(b\) satisfying \(\rho(b)<\rho(f)\) CM, where \(c\in\mathbb{C}\) satisfies \(\Delta_{c}^{n}f\not\equiv 0\), then \(f(z)=a(z)+de^{cz}\), where \(d,c\) are two nonzero constants. In particular, if \(a=b\), then \(a\) reduces to a constant. This result improves and generalizes the recent results of Chen and Chen [3], Liao and Zhang [10] and Lü et al. [11] in a large scale. Also we exhibit some relevant examples to fortify our results.