Uniqueness of Meromorphic Functions with Their Nonlinear Differential Polynomials Share a Small Function

In this paper we deal with the uniqueness of meromorphic functions when two nonlinear differential polynomials generated by two meromorphic functions share a small function. We consider the case for some general differential polynomials [ ( ) n f P f f  ] where P(f) is a polynomial which generalize some result due to Abhijit Banerjee and Sonali Mukherjee [1].


Introduction
In this paper, we use the standard notations and terms in the value distribution theory [2].For any nonconstant meromorphic function   be the set of meromorphic function in the complex plane C which are small functions with respect to f. Set , where a zero point with multiplicity m is counted m times in the set.If these zero points are only counted once, then we denote the set by where a zero point with multiplicity m is counted m times in the set.Let f(z) and g(z) be two transcendental meromorphic functions, If , then we say that ( ) ( ) ( ). a z S f S g   ( ) ( ( ), ) ( ( ), )  E a z f E a z g f z and ( ) g z share the value CM, especially, we say that f(z) and g(z) have the same fixed points when .If ( ) a z ( ) a z z  ( , ) ( , ), E a f E a g  then we say that f(z) and g(z) share the IM.If we say that ( ) a z ( )  ) ) ) ( ),g ( , ( ) Similarly, we have the notations Let f(z) and g(z) be two nonconstant meromorphic the counting function for 1-points of both ( ) f z and ( ) g z about which ( ) f z has larger multiplicity than ( ) g z , with multiplicity is not being counted,  the counting function for common simple 1-points of both f(z) and g(z) where multiplicity is not counted.Similarly, we have the nota- Fang and Fang [3] and in 2004 Lin-Yi [4] in-dependently proved the following result.
Theorem A ( [3,4]).Let f and g be two nonconstant meromorphic functions and be an integer.
( 1 n g g g share 1 CM, then f ≡ g.In 2004 Lin-Yi [5] improved Theorem A by generalizing it in view of fixed point.Lin-Yi [5] proved the following result. Theorem B ([5]).Let f and g be two transcendental meromorphic functions and be an integer.
With the notion of weighted sharing of value recently the first author [6] improved Theorem A as follows.
Theorem C ([6]).Let f and g be two nonconstant meromorphic functions and f g  .In the mean time Lahiri and Sarkar [7] also studied the uniqueness of meromorphic functions corresponding to nonlinear differential polynomials which are different from that of previously mentioned and proved the following.
Theorem D ([7]).Let f and g be two nonconstant meromorphic functions such that , where is an integer then either or .If n is an even integer then the possibility of does not arise.

Theorem E ([1]
).Let f and g be two transcendental meromorphic functions such that   share " ".Then the following holds: , the roots of the equation are distinct and one of f and g is nonentire meromorphic function having only multiple poles, then f ≡ g.
and the roots of the equation coincides, then If n is an even integer then the possibility f g   does not arise.Here, we obtain unicity theorem when [ ( ) n g P g g ] share a small function.Theorem 1.Let f and g be two transcendental meromorphic functions.Let ,  is the first nonzero coefficient from the right, and n, m, k be a positive integer with

Lemmas
In this section we present some lemmas which will be needed in the sequel.Let f, g, F 1 , G 1 be four nonconstant meromorphic functions.Henceforth we shall denote by h and H the following two functions.

Lemma 2.3. ([8]) Let f be a nonconstant meromorphic function and
, where are constants and .Then and 1 ( ) where is a small function of f and g.Then and .
 0, In the same way we can prove , .This proves the Lemma.
where is an integer.
Let be a 1-point of f with multiplicity .Then is a pole of g with multiplicity such that i.e.,

Let
be a zero of with multiplicity 1 .Then 1 is a pole of g with multiplicity , say.So from (2.1) we get 1 z z Sin f ce a pole of ( ) n g P g is either a zero of or a zero of , g we have ,0; ,0; , , 0; N r g denotes the reduced counting functhose zero tion of s of g which are not the zeros of g ( ) P g .
Similarly, we have which is a contradiction.This proves the Lemma.mero-Lemma 2.7.Let f and g be two transcendental morphic function and Similarly, we have Adding (2.5) and (2.6) we obtain , which is a contradiction.So and the Lemma is ed.
a no Lemma 2.9.Let F and be given as in Lemma 2.7 and where are roots of the algebraic equation , ; , ; , ; ,0; ' , .
,0; , where are roots of the algebraic equation Th is proves the Lemma.

Proofs of the Theorems
In a similar manner we can obtain .
for a set of r of finite linear measures.A meromorphic function is called a small function with respect to f(z) k E a z f z a and ( )  g z a have same zeros with the same multiplicities k .Moreover, we also use the following notations.be the corresponding one for which the multiplicity is not counted.Set