A note on entire functions sharing a finite set with their difference operators

Abstract

In this note, we will show that an entire function is equal to its difference operator if it has a growth property and shares a set, where the set consists of two entire functions of smaller orders. This result generalizes a result of Li (Comput.

Uniqueness Problems about Entire Functions with Their Difference Operator Sharing Sets

“…Recently years, Nevanlinna characteristic of f(z + c), the value distribution theory for difference analogue, Nevanlinna theory of the difference operator, and the difference analogue of the lemma on the logarithmic derivative had been built, see e.g., [1][2][3][4][6][7][8][9][10][11][12][13][14]. For meromorphic functions f(z), we define its shift by f c (z) � f(z + c) and its difference operators by Theorem 1.…”

“…Question 2: what will happen if the shift Δ c f(z) be replaced by Δ n c f(z)(n ≥ 2) in eorem 2? Fortunately, we have recently given a positive answer for Question 1 (see [14]). In this work, we also discuss the above problems and especially for Question 2.…”

Entire functions of restricted hyper-order sharing a set of two small functions IM with their linear c-shift operators

“…After publication of Li's [5] result, there was a long gap in research in this direction. Recently in 2019, concerning finite-order entire function, Qi-Wang-Gu [6] removed the restriction ρ(f) ≠ 1 in Theorem B. Not only that, they also ensured the particular form of f in the following manner:…”

“…Theorem C. [6] Let f be a nonconstant entire function with λ(f) < ρ(f) < ∞, let a, b be respectively two distinct entire functions such that ρ(a) < ρ(f), ρ(b) < ρ(f). If f and Δ c f share the set {a, b} CM, then f(z) 5 Ae μz , where A, μ are two nonzero constants satisfying e μc 5 2.…”

A note on entire functions sharing a finite set with applications to difference equations

“…Throughout this paper, we denote by ρ(f ) and λ(f ), the order of f and the exponent of convergence of zeros of f respectively (see [29]): We also need the following notation: Let f be a non-constant meromorphic function, and we define difference operators as…”