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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].

Let

Let f and g be two nonconstant meromorphic functions. Let a be a small function of f and

and g share ¥ CM, if

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

Definition 1.2. 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

where

Remark 1.1. From the above inequalities, we have

Definition 1.3. 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

Remark 1.2. From the above inequality, we have

Definition 1.4. (see [

We write f, g share

R. Bruck [

Theorem A. Let f be a non-constant entire function satisfying

Bruck [

Conjecture 1.1. Let f be a non-constant entire function

Yang [

Theorem B. Let f be a non-constant entire function and

Theorem C. Let f be a non-constant non-entire meromorphic function and

1) f and a have no common poles.

2)

3)

then

In the same paper, Yu [

1) Can a CM shared be replaced by an IM share value?

2) Can the condition

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 [

Theorem D. Let f be a non-constant entire function

Lahiri and Sarkar [

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

2)

3)

In 2005, Zhang [

Theorem F. Let f be a non-constant meromorphic function,

or

or

then

In 2015, Jin-Dong Li and Guang-Xiu Huang proved the following Theorem.

Theorem G. Let f be a non-constant meromorphic function,

or

then

In this paper, we pay our attention to the uniqueness of more generalized form of a function namely

Theorem 1.1. Let f be a non-constant meromorphic function,

or

then

Corollary 1.2. Let f be a non-constant meromorphic function,

or

or

then

Lemma 2.1 (see [

clearly

Lemma 2.2 (see [

where F and G are two non constant meromorphic functions. If F and G share 1 IM and

Lemma 2.3 (see [

be an irreducible rational function in f with constant coefficients

where

Proof of Theorem 1.1. Let

Case 1. Let

By our assumptions, H have poles only at zeros of

here

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

We discuss the following three sub cases.

Sub case 1.1.

Combining (14) and (15), we get

that is

By Lemma 2.1 for

So

which contradicts with (7).

Sub case 1.2.

and

Combining (14) and (17) and (18), we get

that is

By Lemma 2.1 for

So

which contradicts with (8).

Sub case 1.3.

Similarly we have

Combining (14) and (20)-(22), we get

that is

By Lemma 2.1 for

So

which contradicts with (9).

Case 2. Let

on integration we get from (10)

where C, D are constants and

Sub case 2.1. Suppose

and

Since

Suppose

Using the second fundamental theorem for F we get

i.e.,

So, we have

If

and from which we know

We know from (28) that

So from Lemma 2.1 and the second fundamental theorem we get

which is absurd. So

In view of the first fundamental theorem, we get from above

which is impossible.

Sub case 2.2.

If

and

By the second fundamental theorem and Lemma 2.1 for

Hence

So, it follows that

and

This contradicts (7)-(9). Hence

Harina P. Waghamore,S. Rajeshwari, (2015) On Meromorphic Functions That Share One Small Function of Differential Polynomials with Their Derivatives. Applied Mathematics,06,2004-2013. doi: 10.4236/am.2015.612178