Nevanlinna ’ s Five Values Theorem on Annuli

By using the second main theorem of the meromorphic function on annuli, we investigate the problem on two meromorphic functions partially sharing five or more values and obtain some theorems that improve and generalize the previous results given by Cao and Yi.


Introduction and Main Results
The purpose of this paper is to study the uniqueness of two meromorphic functions sharing five or more values.Thus, we always assumed that the reader is familiar with the notations of the Nevanlinna theory, such as T(r, f ), m(r, f ), N(r, f ), and so on (see [1][2][3][4]).We use C to denote the open complex plane, C to denote the extended complex plane and X to denote the subset of C.
In 1929, R. Nevanlinna (see [5]) first investigated the uniqueness of meromorphic functions in the whole complex plane and obtained the well-known theorem: the five I Mtheorem: Theorem 1.1.(see [5]).If f and g are two non-constant meromorphic functions that share five distinct values a 1 , a 2 , a 3 , a 4 , a 5 I M in C, then f (z) ≡ g(z).
After his theorem, there are vast references on the uniqueness of meromorphic functions sharing values and sets in the whole complex plane (see [3]).It is an interesting topic how to extend some important uniqueness results in the complex plane to an angular domain or the unit disc.In the past several decades, the uniqueness of meromorphic functions in the value distribution attracted many investigations.For example, I. Lahiri, H.X. Yi, X.M. Li and A. Banerjee (including [3,[6][7][8]) studied the uniqueness for meromorphic functions on the whole complex plane sharing one, two, three or some sets; M.L. Fang, H.F. Liu, Z.Q.Mao and H.Y. Xu (including [9][10][11]) investigated the shared value of meromorphic functions in the unit disc; J.H. Zheng, Q.C. Zhang, T.B. Cao and W.C. Lin (including [12][13][14][15][16]) considered many uniqueness problem on meromorphic functions on the angular domain.
In 2009, Z.Q.Mao and H.F. Liu [10] gave a different method to investigate the uniqueness problem of meromorphic functions in the unit disc and obtained the following results.Theorem 1.2.(see [10]).Let f , g be two meromorphic functions in D, a j ∈ C(j = 1, 2, . . ., 5) be five distinct values and ∆(θ In the same year, T.B. Cao and H.X. Yi [12] investigated the uniqueness problem of two transcendental meromorphic functions sharing five distinct values in an angular domain and obtained the following theorem: Theorem 1.3.(see [12], Theorem 1.3).Let f and g be two transcendental meromorphic functions.Given one angular domain X = {z : α < arg z < β} with 0 < β − α ≤ 2π, we assume that f and g share five distinct values a j ∈ C(j = 1, 2, 3, 4, 5) I M in X.Then, f (z) ≡ g(z), provided that: where S α,β (r, f ) is used to denote the angular characteristic function of meromorphic function f .
Remark 1.1.We may denote Theorem 1.3 by the five I M theorem in an angular domain.
In 2003, J.H. Zheng [15,16] firstly took into account the value distribution of meromorphic functions in an angular domain.In 2010, J.H. Zheng [17] investigated the uniqueness of the meromorphic function sharing five values in an angular domain, by using Tsjui's characteristic function.
However, the whole complex plane, the unit disc and the angular domain can all be regarded as a simply-connected region; in other words, the theorems stated in the above references are only regarded as the uniqueness results in a simply-connected region.In fact, there exists many other sub-regions in the whole complex plane, such as: the annuli, the m-punctured complex plane, etc.
Recently, there have been some results focusing on the Nevanlinna theory of meromorphic functions on the annulus (see [18][19][20][21][22][23]). The annulus can be regarded as the doubly-connected region.From the doubly-connected mapping theorem [24], we can get that each doubly-connected domain is conformally equivalent to the annulus {z : r < |z| < R}, 0 ≤ r < R ≤ +∞.For two cases: r = 0, R = +∞, simultaneously, and 0 < r < R < +∞; the latter case, the homothety z → z √ rR reduces the given domain to the annulus {z Thus, every annulus is invariant with respect to the inversion z → 1 z in two cases.In 2005, Khrystiyanyn and Kondratyuk [18,19] proposed the Nevanlinna theory for meromorphic functions on annuli (see also [25]).The basic notions of the Nevanlinna theory on annuli will be shown in the next section.Lund and Ye [21] in 2009 studied meromorphic functions on annuli with the form {z : R 1 < |z| < R 2 }, where R 1 ≥ 0 and R 2 ≤ ∞.In 2009 and 2011, Cao [26][27][28] investigated the uniqueness of meromorphic functions on annuli sharing some values and some sets and obtained an analog of Nevanlinna's famous five-value theorem.
Theorem 1.6.(see [26], Theorem 3.2).Let f 1 and f 2 be two transcendental or admissible meromorphic functions on the annulus , where each zero with multiplicity m is counted m times.If we ignore the multiplicity, then the set is denoted by E(a, f ).We use E k) (a, f ) to denote the set of zeros of f − a with multiplicities no greater than k, in which each zero is counted only once.
In this paper, we will further investigate the problem on the five values for meromorphic functions on annuli.To state our main theorem, we first introduce the following definition.
From Theorem 1.7, we can get the following consequences.
Corollary 1.3.Under the assumptions of Corollary 1.2, if E k j ) (a j , f ) = E k j ) (a j , g) and: q then we have f (z) ≡ g(z).

Corollary 1.4. Let f and g be two transcendental or admissible meromorphic functions on the annulus
where γ is stated as in Corollary 1.1; then, f (z) ≡ g(z).
Remark 1.4.If E k j ) (a j , f ) = E k j ) (a j , g) and taking m = 3 in Corollary 1.4, thus Equation (5) becomes: Then, we can get Theorem 1.5 easily.Hence, Theorem 1.7 is an improvement of Theorem 1.5.
Remark 1.5.Throughout our article, we can see that our theorem and corollaries also hold for transcendental meromorphic function in the whole complex plane, which are also extensions of some results given by Nevanlinna, Yi and Cao [3,5,26].

Preliminaries and Some Lemmas
Next, we will introduce the basic notations and conclusion about meromorphic functions on annuli.
For a meromorphic function f on whole plane C, the classical notations of the Nevanlinna theory are denoted as follows: where log + x = max{log x, 0}, and n(t, f ) is the counting function of poles of the function f in {z : |z| ≤ t}.
Let f be a meromorphic function on the annulus A = {z : 1 R 0 < |z| < R 0 }, where 1 < r < R 0 ≤ +∞; the notations of the Nevanlinna theory on annuli will be introduced as follows.Let: where n 1 (t, f ) and n 2 (t, f ) are the counting functions of poles of the function f in {z : t < |z| ≤ 1} and {z : 1 < |z| ≤ t}, respectively.Similarly, for a ∈ C, we have: in which each zero of the function f − a is counted only once.In addition, we use n k) ) to denote the counting function of poles of the function 1 f −a with multiplicities ≤ k (or > k) in {z : t < |z| ≤ 1}, each point counted only once.Similarly, we have the notations N k) The Nevanlinna characteristic of f on the annulus A is defined by: 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 [18]): where (iii) can be called the first fundamental theorem on annuli.
Definition 2.1.Let f (z) be a non-constant meromorphic function on the annulus A = {z : 1 R 0 < |z| < R 0 }, where 1 < R 0 ≤ +∞.The function f is called a transcendental or admissible meromorphic function on the annulus A provided that: or: respectively.
Then, for a transcendental or admissible meromorphic function on the annulus A, S 1 (r, f ) = o(T 0 (r, f )) holds for all 1 < r < R 0 , except for the set r or the set r mentioned in Lemma 2.1, respectively.
The following lemma plays an important role in the proof process of Theorem 1.6, which was given by Cao, Yi and Xu [26].Lemma 2.2.([26], Theorem 2.3) (The second fundamental theorem).Let f be a nonconstant meromorphic function on the annulus A = {z : 1 R 0 < |z| < R 0 }, where 1 < R 0 ≤ +∞.Let a 1 , a 2 , . .., a q be q distinct complex numbers in the extended complex plane C. Then: where S 1 (r, f ) is stated as in Lemma 2.1.
Therefore, we complete the proof of Theorem 1.7.

Definition 1 . 1 .Theorem 1 . 7 .
For B ⊂ A and a ∈ C, we denote by N B 0 (r, 1 f −a ) the reduced counting function of those zeros of f − a on A, which belong to the set B. Let f and g be two transcendental or admissible meromorphic functions on the annulus A