Uniqueness of Meromorphic Functions Whose Differential Polynomials Share One Value

In this paper, we prove a uniqueness theorem of meromorphic functions whose some nonlinear differential shares 1 IM with powers of the meromorphic functions, where the degrees of the powers are equal to those of the nonlinear differential polynomials. This result improves the corresponding one given by Zhang and Yang, and other authors.


Introduction
The meromorphic function mentioned in this paper refers to the meromorphic function over the entire complex plane.Let f and g be two non-constant meromorphic functions.
( ) 0, E ⊂ ∞ means Linear measure finite set.( ) CM is the abbreviation of common multiplicities.And IM is the abbreviation of ignored multiplicities.These concepts can be found in the literature [1].Let a be a finite complex number, if f a − and g a − have the same zero point and the same number of weights, then f and g CM share a.If f a − and g a − have the same zero point without counting the number, then f and g IM share a [2].In addition, the following definitions are required: let p be a positive integer, and ( ( Before, Xiaomin Li and Zhitao Wen expanded Jilong Zhang's theorem, where in this paper, we continuously change , which contained 1 f − .So we expended Xiaomin Li and Zhitao Wen's theorem.
In 2008, Lianzhong Yang and Jilong Zhang proved the following theorems: Theorem B [4] Suppose f is a non-constant meromorphic function, Recently Zhang Jilong improved the above theorem.Get the following result: Theorem C [5] Suppose f is a non-constant entire function, Theorem D [5] Suppose f is a non-constant meromorphic function, Li Xiaomin and Wen Zhitao have improved on the basis of Zhang Jilong's theorem, as follows.
Theorem E [5] Suppose f is a non-constant meromorphic function, k is a positive integer, n is a positive integer and satisfies Theorem F [5] Suppose f is a non-constant meromorphic function, k is a positive integer, n is a positive integer and satisfies Now we mainly improve the theorem of Li Xiaomin.Which that changes n f and . We get the following theorem: Theorem 1 Suppose f is a non-constant meromorphic function, k is a positive integer, n is a positive integer and satisfies ( ) IM share 1, and the zeros of 1 f − with mul- tiplicity 2 at least., then

Some Lemmas
Lemma 1 [6] Suppose F and G are non-constant meromorphic functions, let Lemma 2 [7] Let f be a non-constant meromorphic function, and ( ) , , , , 0 Lemma 3 Let f be a non-constant meromorphic function, ( ) are two positive integers.Let ( ) Proof According to Lemma 2, we obtain It can be seen from the above formula, According to the second basic theorem and (2) , The above formula is combined with (1) to get According to (3) and ( 4), we have According to (2) and ( 5), we can get the conclusion of Lemma 3.
Lemma 4 [8] Let f be a non-constant meromorphic function, , k p are two positive integers.The zero point of Lemma 5 Let f be a non-constant meromorphic function, ( ) , if F and G IM share 1, then Journal of Applied Mathematics and Physics a) Proof According to Lemma 4, we have , This leads to the conclusion (a), obtained from the definition of the 1 , 1 This leads to conclusions (b), the same reason ( ) ( ) and the q form in Lemma 5, we can get (c).
Lemma 6 Suppose F and G are non-constant meromorphic functions, and satisfy ) Let H be defined by Lemma 1.The following two discussions, 6) and ( 8), we get 1 B = .So F G = , contradictory with the assumption of case 1. Journal of Applied Mathematics and Physics Therefore, (8), we get ( ) According to the second basic theorem and (6) (8) (9) (10) we get , , By lemma 2, we have If V was not always equal to 0, (7) can be rewritten into ( ) ( ) Suppose 0 z is a pole of f with multiplicity p, then 0 z is pole of F with mul- tiplicity ( ) . and 0 z is zero of ( ) , , The following two sub-cases are discussed: Sub-case 1.1 suppose If 0 U = , we have where Suppose U is not always equal to 0, let 1 z be a zero of f with multiplicity q, then 1 z is a zero F with multiplicity nq and 1 z is zero of 1 a weight less than p count function of the weight of the value point a within z r < .means the weight of f is not less than p count function of the weight of the value point a within z r < . .See the literature[3] for details.