The Shared Set and Uniqueness of Meromorphic Functions on Annuli Hong

and Applied Analysis 3 Lettingf be ameromorphic function on the annulusA = {z : 1/R 0 < |z| < R 0 }, where 1 < R < R 0 ≤ +∞, the notations of the Nevanlinna theory on annuli had been introduced in [13, 20], such asN 0 (R, f),m 0 (R, f), T 0 (R, f), . . . . In addition, we define Θ 0 (∞, f) = 1 − lim sup R→∞ N 0 (R, f) T 0 (R, f) . (6) We also use nk) 1 (t, 1/(f − a)) (or n(k 1 (t, 1/(f − a))) to denote the counting function of poles of the function 1/(f − a) with multiplicities ≤k (or >k) in {z : t < |z| ≤ 1}, with each point being counted only once. Similarly, we have the notations N k) 1 (t, f),N(k 1 (t, f),Nk) 2 (t, f),N(k 2 (t, f),Nk) 0 (t, f),N(k 0 (t, f). For a nonconstant meromorphic function f on the annulus A = {z : 1/R 0 < |z| < R 0 }, where 1 < R < R 0 ≤ +∞, the following properties will be used in this paper (see [13]): (i) T 0 (R, f) = T 0 (R, 1 f ) , (7) (ii) max{T 0 (R, f 1 ⋅ f 2 ) , T 0 (R, f 1 f 2 ) , T 0 (R, f 1 + f 2 )} ≤ T 0 (R, f 1 ) + T 0 (R, f 2 ) + O (1) , (8)


Introduction
In 1929, Nevanlinna [1] first investigated the uniqueness of meromorphic functions in the whole complex plane and obtained the well-known result-5 IM theorem of two meromorphic functions sharing five distinct values.
The notations of the Nevanlinna theory such as (, ), (, ), and (, ) were usually used in those papers (see [5,9,10]).We use C to denote the open complex plane, C := C ∪ {∞} to denote the extended complex plane, and X to denote the subset of C. Let  be a set of distinct elements in C and where   () = () −  if  ∈ C and  ∞ () = 1/().We also define  X 1 (, ) = ⋃ ∈ { ∈ X : all the simple zeros of   ()} . ( For  ∈ C, we say that two meromorphic functions  and  share the value () in X (or C), if ()− and ()−  have the same zeros with the same multiplicities (ignoring multiplicities) in X (or C).
The whole complex plane C, the unit disc, and angular domain all can be regarded as simply connected regions those results of the uniqueness of shared values and sets in the above cases can also be regarded as the uniqueness of meromorphic functions in simply connected regions.
Thus, it raises naturally an interesting subject on the uniqueness of the meromorphic functions in the multiply connected region.
The main purpose of this paper is to study the uniqueness of meromorphic functions in doubly connected domains of complex plane C. From the doubly connected mapping theorem [11], we can get that each doubly connected domain is conformally equivalent to the annulus { :  < || < }, 0 ≤  <  ≤ +∞.There are two cases: (1)  = 0 and  = +∞ and (2) 0 <  <  < +∞; for case (2) the homothety   → / √  reduces the given domain to the annulus { : 1/ 0 < || <  0 }, where  0 = √/.Thus, every annulus is invariant with respect to the inversion   → 1/ in two cases.The basic notions of the Nevanlinna theory on annuli will be showed in the next section.
In 2012, Cao and Deng [23] investigated the uniqueness of two meromorphic functions in A sharing three or two finite sets; we obtain the following theorems which are an analog of results on C according to Lin and Yi [24].In this paper, we will focus on the uniqueness problem of shared set of meromorphic functions on the annuli.In fact, we will study the uniqueness of meromorphic functions on the annuli sharing one set  = { ∈ A :  1 () = 0}, where and  is a complex number satisfying  ̸ = 0, 1 and we obtain the following results.Theorem 5. Let  and  be two admissible meromorphic functions in the annulus A. If  A (, ) =  A (, ) and  is an integer ≥11, then  ≡ .
A set  is called a unique range set for meromorphic functions on annulus A, if, for any two nonconstant meromorphic functions  and , the condition  A (, ) =  A (, ) implies  ≡ .We denote by ♯ the cardinality of a set .Thus, from Theorem 5, we can get the following corollary.Corollary 6.There exists one finite set  with ♯ = 7, such that any two admissible meromorphic functions  and  on A must be identical if  A (, ) =  A (, ).

Preliminaries and Some Lemmas
Letting  be a meromorphic function on whole plane C, the classical notations of the Nevanlinna theory are denoted as follows: where log +  = max{log , 0} and (, ) is the counting function of poles of the function  in { : || ≤ }.
Khrystiyanyn and Kondratyuk [14] also obtained the lemma on the logarithmic derivative on the annulus A.
In 2005, the second fundamental theorem on the annulus A was first obtained by Khrystiyanyn and Kondratyuk [14].Later, Cao et al. [22] introduced other forms of the second fundamental theorem on annuli as follows.
Remark 16.In fact, from the proof of Theorem 2.3 in [22], under the assumptions of Lemma 15, we can get the following conclusion: where (, ) is stated as in Lemma 15 and  0 0 (, 1/  ) is the counting function for the zeros of   in A, where  does not take one of the values   ( = 1, 2, . . ., ).Definition 17.Let () be a nonconstant meromorphic function on the annulus A = { : 1/ 0 < || <  0 }, where 1 <  0 ≤ +∞.The function  is called an admissible meromorphic function on the annulus A provided that lim sup or lim sup Thus, for an admissible meromorphic function on the annulus A, (, ) = ( 0 (, )) holds for all 1 <  <  0 except for the set Δ  or the set Δ   mentioned in Lemma 13, respectively.
The following result can be derived from the proof of Frank-Reinders' theorem in [25].
Lemma 18.Let  ≥ 6 and Then, () is a unique polynomial for admissible meromorphic functions; that is, for any two admissible meromorphic functions  and  in A, () ≡ () implies  ≡ .
By a similar discussion to the one in [26], one can obtain a stand and Valiron-Mohoko type result in A as follows.
Then, from ( 23) and ( 24), we can get the conclusion of this lemma.
Next, we will give the two main lemmas of this paper as follows.
Thus, we can get a contradiction.Therefore, () ≡ 0; that is, From the above equality, by integration, we can get where , , ,  ∈ C and  −  ̸ = 0.
Case 1 ( ̸ = 0).From the definition of  1 () and (56), we can see that every zero of  1 () + / in A has a multiplicity of at least .Here, three following subcases will be discussed.
Therefore, the proof of Theorem 5 is completed.(66)

The
Then, from Lemma 21, we have  ≡ ( + )/( + ), where , , ,  ∈ C and  −  ̸ = 0. Thus, by using the same argument as that in Theorem 5, we can prove the conclusion of Theorem 7.

Proofs of Theorems 9 and 11
where , , ,  ∈ C and  −  ̸ = 0.By using arguments similar to those in the proof of Theorem 5, we can get that  ≡ .
Therefore, this completes the proof of Theorem 9.  (73)