tailieunhanh - Meromorphic function and its difference operator share two sets with weight k
In this paper, we utilize Nevanlinna value distribution theory to study the uniqueness problem that a meromorphic function and its difference operator share two sets with weight k . Our results extend the previous results. | Turk J Math (2017) 41: 1155 – 1163 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Meromorphic function and its difference operator share two sets with weight k Bingmao DENG, Dan LIU, Degui YANG∗ Institute of Applied Mathematics, South China Agricultural University, Guangzhou, . China Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we utilize Nevanlinna value distribution theory to study the uniqueness problem that a meromorphic function and its difference operator share two sets with weight k . Our results extend the previous results. Key words: Meromorphic function, unicity, shared sets 1. Introduction and notation In this paper, the term ‘meromorphic function’ always means meromorphic in the whole complex plane C . Let f (z) be a nonconstant meromorphic function, and we use the standard notation in Nevanlinna’s theory of meromorphic functions such as T (r, f ), m(r, f ), N (r, f ) , and N (r, f ) (see, ., [2,6]). The notation S(r, f ) is defined to be any quantity satisfying S(r, f ) = o{T (r, f )} as r → +∞ , possibly outside a set of r of finite measure. In addition, we use Nk (r, 1/f ) to denote the counting function for the zeros of f (z) with multiplicity m counted m times if m ≤ k and k + 1 times if m > k . We say that f and g share a CM (IM), if f (z) − a and g(z) − a have the same zeros with the same multiplicities (ignoring multiplicities). We also need the following definitions in this paper. ˆ (= C ∪ {∞}) we denote by Ef (a, k) the set Definition 1 [3] Let k be a positive integer or infinity. For a ∈ C of all a-points of f , where an a-point of multiplicity m is counted m times if m ≤ k and k + 1 times if m > k . ˆ Ef (a, k) = Eg (a, k), we say that f, g Definition 2 [3] Let k be a positive integer or infinity. If for a ∈ C, share the value a with weight k . From these two definitions, we note .
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