Abstract
In this paper, we prove that non-constant meromorphic functions of finite order and their difference operators are identical, if they share four small functions “IM”, or share two small functions and ∞ CM. Our results show that a conjecture posed by Chen–Yi in 2013 is still valid for shared small functions, and improve some earlier results obtained by Li–Yi, Lü et al. We also study the uniqueness of a meromorphic function partially sharing three small functions with their difference operators.
Highlights
1 Introduction and main results In this paper, a meromorphic function always means meromorphic in the complex plane
We adopt the standard notations in Nevanlinna theory; see, e.g. [11, 21]
(r, 1 α dr holds for sufficiently large r and some positive number d
Summary
Theorem A ([5, 18]) If a non-constant meromorphic function f and its derivative f share three distinct finite values a1, a2, a3 IM, f ≡ f. Theorem 1.3 Let f be a non-constant meromorphic function of σ2(f ) < 1, αj ∈ S(f ) (j = 1, 2, 3), and let η be a nonzero finite value. Lemma 2.5 Let f and g be non-constant meromorphic functions, and share four distinct functions αj ∈ S(f ) ∩ S(g) (j = 1, 2, 3, 4) “IM”. If 0, ∞ are the Picard exceptional values of α(z), there exists a non-constant entire function h(z), such that α(z) = eh(z) This implies that T(r, α) ≥ dr – O(1) holds for sufficiently large r and some positive number d. If α(z) has at least one zero or one pole z0, z0 + jη, j ∈ Z are zeros or poles of α(z)
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