eu MULTIPLE VALUES AND UNIQUENESS OF MEROMORPHIC FUNCTIONS ON ANNULI

We assume that the reader is familiar with Nevanlinna’s theory of meromorphic functions (see [13]). The uniqueness of meromorphic functions in the complex plane C is an important subject in the value distribution theory. The uniqueness of meromorphic functions with shared values on C attracted many investigations (see book [13]). Here we shall mainly study the uniqueness of meromorphic functions in doubly connected domains of complex plane C. By Doubly connected mapping theorem [9] 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 7→ z rR reduces the given domain to the annulus


Introduction
We assume that the reader is familiar with Nevanlinna's theory of meromorphic functions (see [13]).The uniqueness of meromorphic functions in the complex plane C is an important subject in the value distribution theory.The uniqueness of meromorphic functions with shared values on C attracted many investigations (see book [13]).Here we shall mainly study the uniqueness of meromorphic functions in doubly connected domains of complex plane C. By Doubly connected mapping theorem [9] 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, in two cases every annulus is invariant with respect to the inversion z → 1 z .
The uniqueness theory of meromorphic function is an interesting problem, recently Khrystiyanyn and Kondratyuk [6,7 ] proposed the Nevenlinna theory of meromorphic functions on annuli, see also an important paper [3].We will show the basic notions of the Nevanlinna theory on annuli in the next section.In this paper, we mainly study the uniqueness problem of meromorphic functions on annuli, and extend some uniqueness theorems of meromorphic functions dealing with multiple values and deficient values to meromorphic functions on annili.

Basic Notations in the Nevanlinna Theory on Annuli
Let f be a meromorphic function on the annulus A = z : 1 R 0 < |z| < R 0 .We recall classical notations of 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}.Here we 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 the poles of the function f in {z : t < |z| ≤ 1} and {z : 1 < |z| ≤ t}, respectively.The Nevanlinna charecteristic of f on the annulus A is defined by Definition .[4] Let f (z) be a non-constant meromorphic function on the annulus A(R 0 ) = {z : 1/R 0 < |z| < R 0 }, where 1 < R 0 < +∞.The function f is called a transcedental or admissible meromorphic function on the annulus respectively.
Thus for a transcedental 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 △ R or the set △ ′ R mentioned in Theorem 2.1, respectively.Definition .Let f (z) be a meromorphic functions on the annulus A = z : 1 R 0 < |z| < R 0 , where 1 < R < R 0 ≤ +∞.Let a be any arbitrary complex number.The Valiron deficiency of f (z) on the annulus A with respect to the value ′ a ′ will be defined by 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 where to denote the counting function of poles of the function 1  f −a with the multiplicities ≤ k (or > k) in {z : t < |z| ≤ 1}, each point counted only once.Similarly, we can give the notations Let ′ a ′ be an arbitrary complex number, and k be a positive integer.Then Theorem 2.1.

Multiple Values and Uniqueness of Meromorphic Functions on Annuli
Let f be a meromorphic function on the annulus A = z : 1 R 0 < |z| < R 0 , where 1 < R < R 0 ≤ +∞, and ′ a ′ be a complex number in the extende complex plane C = C ∪ {∞}.Write E(a, f ) = {z ∈ A : f (z) − a = 0}, 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.
We now show our main results below which is an analog of a result on the plane C obtained by H. X. Yi [12](see Theorem 3.19 and 3.20 in [13]).Theorem 3.1.Let f 1 and f 2 be two transcedental or admissible meromorphic functions on the annulus A = z : 1 R 0 < |z| < R 0 , where 1 < R < R 0 ≤ +∞.Let a j (j = 1, 2, ..., q) be q distinct complex numbers in C, and k j (j = 1, 2, ..., q) be positive integers or ∞ satisfying and Set If and then f 1 (z) ≡ f 2 (z).