On Entire and Meromorphic Functions That Share One Small Function with Their Differential Polynomial

In this paper, a meromorphic functions mean meromorphic in the whole complex plane.We use the standard notations of Nevanlinna theory (see [1]). A meromorphic function a(z) is called a small function with respect to f(z) if T(r, a) = S(r, f), that is, T(r, a) = o(T(r, f)) as r → ∞ possibly outside a set of finite linearmeasure. Iff(z)−a(z) and g(z)−a(z) have the same zeros with samemultiplicities (ignoringmultiplicities), then we say thatf(z) and g(z) share a(z)CM (IM). For any constant a, we denote by Nk)(r, 1/(f − a)) the counting function for zeros of f(z) − a with multiplicity no more than k and Nk)(r, 1/(f − a)) the corresponding for which multiplicity is not counted. LetN(k(r, 1/(f−a)) be the counting function for zeros of f(z) − a with multiplicity at least k andN(k(r, 1/(f − a)) the corresponding for which the multiplicity is not counted. Let f and g be two nonconstant meromorphic functions sharing value 1 IM. Let z0 be common one point of f and g with multiplicity p and q, respectively. We denote byNL(r, 1/(f − 1)) (NL(r, 1/(f − 1))) the counting (reduced) function of those 1 points of f where p > q; byN E (r, 1/(f −


Introduction and Main Results
In this paper, a meromorphic functions mean meromorphic in the whole complex plane.We use the standard notations of Nevanlinna theory (see [1]).A meromorphic function () is called a small function with respect to () if (, ) = (, ), that is, (, ) = ((, )) as  → ∞ possibly outside a set of finite linear measure.If ()−() and ()−() have the same zeros with same multiplicities (ignoring multiplicities), then we say that () and () share () CM (IM).
For any constant , we denote by  ) (, 1/( − )) the counting function for zeros of () −  with multiplicity no more than  and  ) (, 1/( − )) the corresponding for which multiplicity is not counted.Let  ( (, 1/( − )) be the counting function for zeros of () −  with multiplicity at least  and  ( (, 1/( − )) the corresponding for which the multiplicity is not counted.

Conjecture 1.
Let  be a nonconstant entire function such that the hyper-order  2 () of  is not a positive integer and  2 () < ∞.If  and   share a finite value  CM, then (  − )/( − ) = , where  is a nonzero constant.
In [3], under an additional hypothesis, Brück proved that the conjecture holds when  = 1.
Many people extended this theorem and obtained many results.In 2003, Yu [4] proved the following theorem.
In the same paper, the author posed the following questions.In 2004, Liu and Gu [5] applied different method and obtained the following theorem which answers some questions posed in [4].
Recently, Zhang and Lü [6] considered the problem of meromorphic functions sharing one small function with its th derivative and proved the following theorem.
Regarding these results, a natural question is what can be said when a nonconstant meromorphic function  shares one nonzero small meromorphic function () with [], where [] is a differential polynomial in .
we get

Proof of Theorems
Proof of Theorem 3.
From ( 17) and (28), we have where  0 (, 1/  ) denotes the counting function corresponding to the zeros of   which are not the zeros of  and  − 1.
Proof of Theorem 7.  is a nonconstant entire function.Taking (, ) = 0 in proof of Theorem 3, we obtain Theorem 7.
Proof of Theorem 9.  is a nonconstant entire function.Taking (, ) = 0 in proof of Theorem 5, we obtain Theorem 9.