On Meromorphic Functions That Share One Small Function of Differential Polynomials with Their Derivatives

In this paper, we study the problem of meromorphic functions that share one small function of differential polynomial with their derivatives and prove one theorem. The theorem improves the results of Jin-Dong Li and Guang-Xin Huang [1].


Introduction and Results
Let  denote the complex plane and f be a nonconstant meromorphic function on  .We assume the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as ( ) ( ) , , , , T r f m r f ( ) , N r f (see, e.g., [2]  g We say that f, g share a counting multiplicities (CM) if , f a g a − − have the same zeros with the same multiplicities and we say that f, g share a ignoring multiplicities (IM) if we do not consider the multiplicities.In addition, we say that f and g share ∞ CM, if 1 1 , f g share 0 CM, and we say that f and g share ∞ IM, if 1 1 , f g share 0 IM.Suppose that f and g share a IM.Throughout this paper, we denote by 1 , 1 the reduced counting function of H. P. Waghamore, S. Rajeshwari 2005 those common a-points of f and g in z r < , where the multiplicity f each a-point of f is greater than that of the corresponding a-point of g, and denote by 11 1 , N r f a the counting function for common simple 1-point of both f and g, and the counting function of those 1-points of f and g where 2 p q = ≥ .In the same way, we can define (2 11 1 1 , , , 1 1 If f and g share 1 IM, it is easy to see that In addition, we need the following definitions: Definition 1.1.Let f be a non-constant meromorphic function, and let p be a positive integer and we denote the counting function of those a-points of f (counted with proper multiplicities) whose multiplicities are not greater that p, by ) , , , , Let f be a non-constant meromorphic function, and let a be any value in the extended complex plane, and let k be an arbitrary nonnegative integer.We define From the above inequalities, we have ( ) ( ) ( ) ( ) Let f be a non-constant meromorphic function, and let a be any value in the extended complex plane, and let k be an arbitrary nonnegative integer.We define ( ) ( ) From the above inequality, we have , a p for all integers p with 0 p k ≤ ≤ .Also, we note that f, g share a value a IM or CM if and only if they share ( ) ,0 a or ( ) , a ∞ , respectively.R. Bruck [5] first considered the uniqueness problems of an entire function sharing one value with its derivative and proved the following result.
Theorem A. Let f be a non-constant entire function satisfying . If f and f ′ share the value 1 CM, then 1 1 for some nonzero constant c.
Bruck [5] further posed the following conjecture.Conjecture 1.1.Let f be a non-constant entire function ( ) Yang [6] proved that the conjecture is true if f is an entire function of finite order.Yu [7] considered the problem of an entire or meromorphic function sharing one small function with its derivative and proved the following two theorems.
Theorem B. Let f be a non-constant entire function and Theorem C. Let f be a non-constant non-entire meromorphic function and f and a have no common poles.
2) f a − and ( ) where k is a positive integer.In the same paper, Yu [7] posed the following open questions. 1) Can a CM shared be replaced by an IM share value?
2) Can the condition ( ) of theorem B be further relaxed?
3) Can the condition 3) in theorem C be further relaxed?4) Can in general the condition 1) of theorem C be dropped?In 2004, Liu and Gu [8] improved theorem B and obtained the following results.Theorem D. Let f be a non-constant entire function Lahiri and Sarkar [9] gave some affirmative answers to the first three questions improving some restrictions on the zeros and poles of a.They obtained the following results.
Theorem E. Let f be a non-constant meromorphic function, k be a positive integer, and 1) a has no zero (pole) which is also a zero (pole) of f or ( ) [10] improved the above results and proved the following theorems.
Theorem F. Let f be a non-constant meromorphic function, ( ) ( ) Theorem G. Let f be a non-constant meromorphic function, ( ) ( ) or 0 l = and .

Lemmas
Lemma 2.1 (see [1]).Let f be a non-constant meromorphic function, , k p be two positive integers, then where F and G are two non constant meromorphic functions.If F and G share 1 IM and 0 H ≡ / , then ( ) ( ) ( ) [11]).Let f be a non-constant meromorphic function and let ( ) .
Then F and G share ( ) 1,l , except the ze- ros and poles of ( ) a z .Let H be defined by (10).
Case 1.Let 0. H ≡ / By our assumptions, H have poles only at zeros of F ′ and G′ and poles of F and G, and those 1-points of F and G whose multiplicities are distinct from the multiplicities of corresponding 1-points of G and F respectively.Thus, we deduce from (10) that ( ) ( ) is the counting function which only counts those points such that 0 F ′ = but ( ) Because F and G share 1 IM, it is easy to see that By the second fundamental theorem, we see that Using Lemma 2.2 and ( 11), ( 12) and ( 13), we get 1 We discuss the following three sub cases.Sub case 1.1.
( ) ( ) ( ) Similarly we have , , Combining ( 14) and ( 20)-( 22), we get ( ) ( ) ( ) , , , By Lemma 2.1 for 2 p = and for 1 p = respectively, we get ( ) From ( 7)-( 9) we know respectively and from which we know which is impossible.Sub case 2.2.0 D = and so from (24) we get ( )  By the second fundamental theorem and Lemma 2.1 for 1 p = and Lemma 2.3 we have , [3]), and ( ) , S r f denotes any quantity that satisfies the condition f = as r → ∞ outside of a possible exceptional set of finite linear measure.A meromorphic function a is called a small function with respect to f, provided that ( ) ( ) , , T r a S r f = .Let f and g be two nonconstant meromorphic functions.Let a be a small function of f and .
Jin-Dong Li and Guang-Xiu Huang proved the following Theorem.

Theorem 1 . 1 .
In this paper, we pay our attention to the uniqueness of more generalized form of a function namely Let f be a non-constant meromorphic function, ( ) ( )

Definition 1.4. (see [4]). Let
k be a nonnegative integer or infinity.For a C ∈ we denote by In view of first fundamental theorem, we get from above