The Uniqueness of Analytic Functions on Annuli Sharing Some Values

and Applied Analysis 3 where log x max{logx, 0} and n t, f is the counting function of poles of the function f in {z : |z| ≤ t}. We here show the notations of the Nevanlinna theory on annuli. Let


Introduction
In this paper, we will study the uniqueness problem of analytic functions in the field of complex analysis and adopt the standard notations of the Nevanlinna theory of meromorphic functions as explained see 1-3 . We use C to denote the open complex plane, C to denote the extended complex plane, and X to denote the subset of C. For a ∈ C, we say that f z − a and g z − a have the same zeros with the same multiplicities ignoring multiplicities in X or C if two meromorphic functions f and g share the value a CM IM in X or C . In addition, we also use f a g a in X or C to express that f and g share the value a CM in X or C , f a ⇔ g a in X or C to express that f and g share the value a IM in X or C , and f a ⇒ g a in X or C to express that f a implies g a in X or C .
In 1929, Nevanlinna see 4 proved the following well-known theorem.
Theorem 1.1 see 4 . If f and g are two nonconstant meromorphic functions that share five distinct values a 1 , a 2 , a 3 , a 4 , and a 5 IM in C, then f z ≡ g z .

Abstract and Applied Analysis
After his theorem, the uniqueness theory of meromorphic functions sharing values in the whole complex plane attracted many investigations see 2 . In 2003, Zheng 5 studied the uniqueness problem under the condition that five values are shared in some angular domain in C. There were many results in the field of the uniqueness with shared values in the complex plane and angular domain, see 5-12 . The whole complex plane C and angular domain all can be regarded as simply connected region. Thus, it is interesting to consider the uniqueness theory of meromorphic functions in the multiply connected region. Here, we will mainly study the uniqueness of meromorphic functions in doubly connected domains of complex plane C. By the doubly connected mapping theorem 13 each doubly connected domain is conformally equivalent to the annulus {z : r < |z| < R}, 0 ≤ r < R ≤ ∞. We consider only two cases: r 0, R ∞ simultaneously and 0 < r < R < ∞. In the latter case the homothety z → z/ √ rR reduces the given domain to the annulus {z : 1/R 0 < |z| < R 0 }, where R 0 R/r. Thus, every annulus is invariant with respect to the inversion z → 1/z in two cases.
In 2005, Khrystiyanyn and Kondratyuk 14,15 proposed the Nevanlinna theory for meromorphic functions on annuli see also 16 . We will show the basic notions of the Nevanlinna theory on annuli in the next section.  In fact, we will prove some general theorems on the uniqueness of analytic functions on the annuli sharing four values in this paper see Section 3 , and these theorems improve Theorem 1.4.

Basic Notions in the Nevanlinna Theory on Annuli
Let f be a meromorphic function on the annulus A {z : 1/R 0 < |z| < R 0 }, where 1 < R < R 0 ≤ ∞. We recall the classical notations of the Nevanlinna theory 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}. We here show the notations of the Nevanlinna theory on annuli. 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. The Nevanlinna characteristic of f on the annulus A is defined by and has the following properties.

Proposition 2.1 see 14 . Let f be a nonconstant meromorphic function on the annulus
By Proposition 2.1, the first fundamental theorem on the annulus A is immediately obtained.

Theorem 2.2 see 14 the first fundamental theorem . Let f be a nonconstant meromorphic function on the annulus
for every fixed a ∈ C.
Khrystiyanyn and Kondratyuk also obtained the lemma on the logarithmic derivative on the annulus A.

Theorem 2.3 see 15 lemma on the logarithmic derivative . Let f be a nonconstant meromorphic function on the annulus
We denote the deficiency of a ∈ C C ∪ {∞} with respect to a meromorphic function f on the annulus A by and denote the reduced deficiency by in which each zero of the function f − a is counted only once. In addition, we use n  a 1 , a 2 , . . . , a q be q distinct complex numbers in the extended complex plane C. Let λ ≥ 0. Then, Abstract and Applied Analysis 5 and (i) in the case R 0 ∞, Definition 2.5. Let f z be a nonconstant meromorphic function on the annulus A {z : The function f is called a transcendental or admissible meromorphic function on the annulus A provided that lim sup Thus, for a transcendental or admissible meromorphic function on the annulus A, S R, f o T 0 R, f holds for all 1 < R < R 0 except for the set Δ R or the set Δ R mentioned in Theorem 2.3, respectively.

The Main Theorems and Some Lemmas
Now we show our main results, which improve Theorem 1.4.  To prove the above theorems, we need some lemmas as follows.

Theorem 3.1. Let f, g be two analytic functions on the annulus
Lemma 3.4. Let f, g be two distinct analytic functions on the annulus A {z : 1/R 0 < |z| < R 0 }, where 1 < R 0 ≤ ∞, and let a j ∈ C j 1, 2, 3, 4 be four distinct complex numbers. If f a j ⇒ g a j in A for j 1, 2, 3, 4 and if f is transcendental or admissible on A, then g is also transcendental or admissible.
Proof. By the assumption of Lemma 3.4 and applying Theorem 2.4 ii , we can get

3.2
Therefore holds for all 1 < R < R 0 except for the set Δ R or the set Δ R mentioned in Theorem 2.3, respectively. Then, from Definition 2.5, we get that g is transcendental or admissible on A.

Lemma 3.5. Suppose that f is a transcendental or admissible meromorphic function on the annulus
Let P f a 0 f p a 1 f p−1 · · · a p a 0 / 0 be a polynomial of f with degree p, where the coefficients a j j 0, 1, . . . , p are constants, and let b j j 1, 2, . . . , q be q q ≥ p 1 distinct finite complex numbers. Then, Proof. From Theorem 2.3 and the definition of m 0 R, f , transcendental and admissible function, we can get this lemma by using the same argument as in Lemma 4.3 in 2 . that f and g share a 1 , a 2 IM in A, and f a 3 ⇒ g a 3 in A and f a 4 ⇒ g a 4 in A, and a j ∈ C j 1, 2, 3, 4 are four distinct finite complex numbers. If f is a transcendental or admissible function on A, then g is also transcendental or admissible, and

Lemma 3.6. Let f, g be two distinct analytic functions on the annulus
Abstract and Applied Analysis where S R : S R, f S R, g .
Proof. By the assumption of this lemma and by Theorem 2.4 ii , we have T 0 R, f ≤ 3T 0 R, g S R, f and T 0 R, g ≤ 3T 0 R, f S R, g . Thus, we can get S R, f S R, g . Let From the conditions of this lemma, we can get that η is analytic on A and η / ≡ 0 unless f ≡ g. By Lemma 3.5, we have m 0 R, η S R, f S R, g S R . Thus, we can get S R, η S R .
Since f, g are two nonconstant analytic functions on annulus A and share a 1 , a 2 IM in A and f a 3 ⇒ g a 3 and f a 4 ⇒ g a 4 in A, again by Theorem 2.4, we have S R, g , 3.9 From 3.8 and 3.11 , we can get i , and from 3.7 , 3.8 , and i , we can get ii , and from 3.6 , 3.8 , 3.10 , 3.11 , and i , we can get iii . Thus, we can deduce that iv and v hold easily from 3.6 -3.11 and i -iii . Now, we will prove that vi holds as follows.
First, we can rewrite 3.5 as 3.12 From 3.12 and Lemma 3.5, we can get m 0 R, f ≤ m 0 R, f S R, f . Since f is analytic on A, we have

Abstract and Applied Analysis
From the fact that f is transcendental or admissible, we have S R, f . Thus, we can get From 3.14 , 3.15 and the fact that f is transcendental or admissible, we can get Similarly, we can get T 0 R, g T 0 R, g S R, g . Thus, we complete the proof of this lemma.

4.5
From 4.5 and Lemma 3.6 iv , we have T 0 R, f ≤ S R . Thus, since f, g are transcendental or admissible functions on A, that is, f and g are of unbounded characteristic, and from the definition of S R , we can get a contradiction. Assume that one of ψ 1 and ψ 2 is identically zero, say ψ 1 ≡ 0; then we have From 3.5 , we can see that g z 1 a 4 implies that f z 1 a 4 for such z 1 ∈ A satisfying η z 1 / 0. Since T 0 R, η S R , we have From 4.6 and 4.7 , we can get Similarly, when ψ 2 ≡ 0, we can get From 4.8 , 4.9 , and Lemma 3.6 i , v , we can get Since f, g are transcendental or admissible functions on the annulus A, we can get a contradiction again. Thus, we complete the proof of Theorem 3.1.

The Proof of Theorem 3.2
Suppose that f / ≡ g. By Theorem 2.4 ii and the fact that f is transcendental or admissible on A, we have S R, f

5.1
Therefore, we have ≤ T 0 R, g S R, f S R, g .

5.2
Similarly, we have From 5.2 and 5.3 , we can see that T 0 R, f T 0 R, g S R, f S R, g , and S R, f S R, g , N 0 R, 1 g − a 4 S R, f S R, g .

5.4
Thus, from 5.2 , 5.3 , and the definition of S R , we can get that g is also transcendental or admissible on A when f is transcendental or admissible on A.
From 5.1 -5.4 , we can also get From 5.5 , we can see that "almost all" of zeros of f − a i i 1, 2 in A are simple. Similarly, "almost all" of zeros of g − a i i 1, 2 in A are simple, too. Let g − a 1 g − a 4 , f − a 2 f − a 3 − a 2 − a 4 g g − a 1 g − a 2 g − a 4 .