ABSTRACT

In this paper, we shall study the uniqueness problems of meromorphic functions of differential polynomials sharing two values IM. Our results improve or generalize many previous results on value sharing of meromorphic functions.

1. Introduction

Let and be two non-constant meromorphic functions defined in the open complex plane. Let, we say that and share CM (counting multiplicities) if, have the same zeros with the same multiplicities and we say that and share (ignoring multiplicities) if we do not consider the multiplicities. We denote by the Nevanlinna characteristic function

of the meromorphic function and by any quantity satisfying as

possibly outside a set of finite linear measure. denotes the truncated counting function bounded by. Moreover, denotes the greatest common divisor of positive integers.

For the sake of simplicity, let be a nonnegative integer, be complex constants. Define

(1.1)

In 1929, Nevanlinna [1] proved the following well-know result which is the so called Nevanlinna five values theorem.

Theorem A Let and be two non-constant meromorphic functions. If and share five distinct values IM, then.

Moreover, he got.

Theorem B Let and be two distinct non-constant meromorphic functions and be four distinct values. If and share CM, then is a Mobius transformation of.

In 1976, L. Rubel asked the following question:

Whether CM can be replaced by IM in the hypothesis of Theorem A with the same conclusion or not?

In 1979, G. G. Gundersen [2] gave a negative answer for this question by the following counterexample:

,

where is a non-constant entire function. It is easy to verify that and share the four values, where none of the four values are shared CM, and is not a Mobius transformation of.

On the other hand, G. G. Gundersen [3] proved the following result which is an improvement of Theorem B. Theorem C. If two distinct non-constant meromorphic functions share two values CM and share two other values IM, then the functions share all four values CM (hence the conclusions of Theorem B hold).

In this paper, we shall show that similar conclusions hold for certain types of differential polynomials when they share two values IM.

Theorem 1.1 Let and be two non-constant meromorphic functions, , and be

three integers with and be defined as in (1.1). If and

share 1 and IM, then

1) when,;

2) when, one of the following two cases holds:

3) for a constant such that,

4), where and are three constants satisfying

.

Remark 1.1 “and share IM” and share IM”. Moreover,

from, one cannot get for some constant. For example, let,

, then where is a non-constant meromorphic function. Obviously, for some canstant but.

Now we give some corollaries of Theorem 1.1. Corollary 1.2 and Corollary 1.3 improve Theorems D and E, respectively.

Corollary 1.2 Let and be two non-constant meromorphic functions, and let be two positive

integers with. If and share 1 IM, and share IM, then either, where and are three constants satisfying, or

for a constant such that.

Corollary 1.3 Let and be two non-constant meromorphic functions satisfying, and let be two positive integers with. If and share 1 IM,

and share IM, then.

Corollary 1.4 Let and be two non-constant meromorphic functions, and let be two positive integers with, be a nonzero constant. If and

share 1 IM, and share IM, then for some constant such that, where

.

Theorem 1.1 generalizes the following result that was obtained by Zhang, Chen and Lin [4].

Theorem D Let and be two non-constant entire functions. Let, and be three positive integ-

ers with and let or, where are complex constants. If and share 1 CM, then

1) when, either for a constant such that, where, for some, or and satisfy the algebraic

equation,

where;

2) when, either, where and are three constants satisfying, or for a constant such that.

Corollaries 1.2-1.4 greatly improve the following result that was obtained by Liu [5] by reducing the lower bound of. Moreover, the proofs of Corollaries 1.2 - 1.4 fill some gaps appeared in the proof of Theorem E.

Theorem E Let and be two non-constant meromorphic functions, and let, and be three

positive integers with, and, be two constants such that. If

and share 1 IM, and share IM, then\\

1) when, If and, then.

If and, then;

2) when, if and, then either, where is a constant satisfying, or, where and are three constants satisfying

or Here, , where

if, if.

2. Preliminary Lemmas

Let

(2.1)

(2.2)

where and are meromorphic functions.

Lemma 2.1 [6] Let be a non-constant meromorphic function and let be

small functions with respect to. Then

Lemma 2.2 [7] Let be a non-constant meromorphic function, be two positive integers. Then

Lemma 2.3 [8-10] Let be a non-constant meromorphic function, and let be a positive integer. Suppose that, then

By using the similar method to Banerjee [11, Lemma 2.14], we can prove the following Lemma.

Lemma 2.4 Let, and be defined as in (2.1). If and share 1 CM and IM, and, then, and

the same inequality holding for.

Lemma 2.5 [12] Let, and be defined as in (2.2). If and share IM, and, then.

Lemma 2.6 [13] If and share 1 IM, then

.

Lemma 2.7 Let, be two non-constant meromorphic functions, be defined as in (2.2), where

, , is defined as in (1.1), , and are three in-

tegers. If, and share 1 CM and IM, then

(2.3)

Proof Since, and share IM, suppose that is a pole of with multiplicity, a pole of with multiplicity, then is a pole of with multiplicity, a pole of with

multiplicity, thus is a zero of with multiplicity

, and is a zero of

with multiplicity, hence is a zero of with multiplicity at least

. So

(2.4)

By the logarithmic derivative lemma, we have. Note that and share 1

IM, by Lemma 2.6, so we have

(2.5)

From (2.4) and (2.5) we get (2.3). This proves Lemma 2.7.

Lemma 2.8 [14] Let and be two non-constant meromorphic functions, and be two

positive integers. If, then for a constant such that.

By the same reason as in Lemma 5 of [8], we obtain the following lemma.

Lemma 2.9 Let and be two non-constant meromorphic functions. Let be defined as in (1.1),

and, and be three  integers with. If, then.

Lemma 2.10 [15] Let and be non-constant meromorphic functions, be two positive integers with, and let be defined as in (1.1), be a small function with respect to

with finitely many zeros and poles. If, and share IM, then

is reduced to a nonzero monomial.

Use the proof of Theorem 3 in [15] and we obtain.

Lemma 2.11 Let and be non-constant meromorphic functions, be two positive integers with

. If, and share IM, then, where and are three constants satisfying.

Lemma 2.12 [16] Let and are relatively prime integers, and let be a complex number such that. Then there exists one and only one common zero of and.

3. Proof of Theorem 1.1

Let, , , , then and share 1 IM and

IM. Suppose that, then, and.

Case 1.. By Lemma 2.4 we have

(3.1)

By Lemma 2.2 with, we obtain

(3.2)

and

(3.3)

Combining (3.1) - (3.3) gives

It follows from Lemma 2.1 and the above inequality that

(3.4)

Similarly we have

(3.5)

Note that .. From (3.4) and (3.5) we deduce that

. (3.6)

Note that and we get (2.3). By Lemma 2.2 with, we obtain

(3.7)

and

(3.8)

From (2.3), (3.7) and (3.8) we get

(3.9)

Combining (3.6) - (3.9) gives

(3.10)

which is a contradiction since. Thus. Similar to the proof of [17, Lemma 3], we obtain

1), or

2).

By Lemma 2.10, the case of 1) is impossible. By Lemma 2.9, we get from 2).

Case 2.. Similar to the proof of Case 1, we get

, (3.11)

which is a contradiction since. Thus. and we have

3), or

4).

For 3), by Lemma 2.11, we get, where and are three con-

stants satisfying.

For 4), By Lemma 2.8, we get for a constant such that. This completes the proof of Theorem 1.1.

4. Proof of Corollaries 1.2 - 1.4

The proof of Corollary 1.2 is the same to the proof of Case 2 of Theorem 1.1, we only need to let. Thus we omit the proof here.

Now we prove Corollary 1.3, Let, similar to (3.10), we get

, (3.12)

which is a contradiction since. Thus and we have

1), or

2).

By Lemma 2.10, the case of (i) is impossible. By Lemma 2.9, we get from 2).

Similar to the proof of Theorem 2 in [14], we get. This proves Corollary 1.3.

Next we prove Corollary 1.4.

According to the proof of Case 1 in Theorem 1.1, we have

1), or

2).

By Lemma 2.10, the case of 1) is impossible. By Lemma 2.9, we get from 2).

Let. If is not a constant, then substitute into and we get

where are distinct roots of the algebraic equation, are distinct roots of the algebraic equation.

Suppose that, then, , where, are co-prime integers and,

thus, which implies. By Lemma 2.12, there exists one and only one common zero of and, namely. Therefore, there exists at least of different from. Suppose that are different from, then all zeros of have order of at least m. Applying the second fundamental theorem to gives

Note that and we get a contradiction. Thus is a constant. From (4.2) we have and, thus for some constant such that, where. This proves Corollary 1.4.

5. Open Problem

For further study, we pose the following. Problem: What form of implies for some constant?

Let and be two non-constant meromorphic functions defined in the open complex plane. Let, we say that and share CM (counting multiplicities) if, have the same zeros with the same multiplicities and we say that and share (ignoring multiplicities) if we do not consider the multiplicities. We denote by the Nevanlinna characteristic function of the meromorphic function and by any quantity satisfying as

possibly outside a set of finite linear measure. denotes the truncated counting function bounded by. Moreover, denotes the greatest common divisor of positive integers.

For the sake of simplicity, let be a nonnegative integer, be complex constants. Define

(1.1)

In 1929, Nevanlinna [1] proved the following well-know result which is the so called Nevanlinna five values theorem.

Theorem A Let and be two non-constant meromorphic functions. If and share five distinct values IM, then.

Moreover, he got.

Theorem B Let and be two distinct non-constant meromorphic functions and be four distinct values. If and share CM, then is a Mobius transformation of.

In 1976, L. Rubel asked the following question:

Whether CM can be replaced by IM in the hypothesis of Theorem A with the same conclusion or not?

In 1979, G. G. Gundersen [2] gave a negative answer for this question by the following counterexample:

where is a non-constant entire function. It is easy to verify that and share the four values, where none of the four values are shared CM, and is not a Mobius transformation of.

On the other hand, G. G. Gundersen [3] proved the following result which is an improvement of Theorem B. Theorem C. If two distinct non-constant meromorphic functions share two values CM and share two other values IM, then the functions share all four values CM (hence the conclusions of Theorem B hold).

In this paper, we shall show that similar conclusions hold for certain types of differential polynomials when they share two values IM.

Theorem 1.1 Let and be two non-constant meromorphic functions, , and be three integers with and be defined as in (1.1). If and

share 1 and IM, then 1) when,;

2) when, one of the following two cases holds:

3) for a constant such that4), where and are three constants satisfying

.

Remark 1.1 “and share IM” and share IM”. Moreoverfrom, one cannot get for some constant. For example, let,

, then where is a non-constant meromorphic function. Obviously, for some canstant but.

Now we give some corollaries of Theorem 1.1. Corollary 1.2 and Corollary 1.3 improve Theorems D and E, respectively.

Corollary 1.2 Let and be two non-constant meromorphic functions, and let be two positive integers with. If and share 1 IM, and share IM, then either, where and are three constants satisfying, or

for a constant such that.

Corollary 1.3 Let and be two non-constant meromorphic functions satisfying, and let be two positive integers with. If and share 1 IM,

and share IM, then.

Corollary 1.4 Let and be two non-constant meromorphic functions, and let be two positive integers with, be a nonzero constant. If and

share 1 IM, and share IM, then for some constant such that, where

.

Theorem 1.1 generalizes the following result that was obtained by Zhang, Chen and Lin [4].

Theorem D Let and be two non-constant entire functions. Let, and be three positive integers with and let or, where are complex constants. If and share 1 CM, then 1) when, either for a constant such that, where, for some, or and satisfy the algebraic equationwhere;

2) when, either, where and are three constants satisfying, or for a constant such that.

Corollaries 1.2-1.4 greatly improve the following result that was obtained by Liu [5] by reducing the lower bound of. Moreover, the proofs of Corollaries 1.2 - 1.4 fill some gaps appeared in the proof of Theorem E.

Theorem E Let and be two non-constant meromorphic functions, and let, and be three positive integers with, and, be two constants such that. If

and share 1 IM, and share IM, then\\

1) when, If and, then.

If and, then;

2) when, if and, then either, where is a constant satisfying, or, where and are three constants satisfying

or Here, , where

if, if.

2. Preliminary Lemmas

Let

(2.1)

(2.2)

where and are meromorphic functions.

Lemma 2.1 [6] Let be a non-constant meromorphic function and let be small functions with respect to. Then

Lemma 2.2 [7] Let be a non-constant meromorphic function, be two positive integers. Then

Lemma 2.3 [8-10] Let be a non-constant meromorphic function, and let be a positive integer. Suppose that, then

By using the similar method to Banerjee [11, Lemma 2.14], we can prove the following Lemma.

Lemma 2.4 Let, and be defined as in (2.1). If and share 1 CM and IM, and, then, and

the same inequality holding for.

Lemma 2.5 [12] Let, and be defined as in (2.2). If and share IM, and, then.

Lemma 2.6 [13] If and share 1 IM, then

.

Lemma 2.7 Let, be two non-constant meromorphic functions, be defined as in (2.2), where

, , is defined as in (1.1), , and are three integers. If, and share 1 CM and IM, then

(2.3)

Proof Since, and share IM, suppose that is a pole of with multiplicity, a pole of with multiplicity, then is a pole of with multiplicity, a pole of with multiplicity, thus is a zero of with multiplicity

, and is a zero of

with multiplicity, hence is a zero of with multiplicity at least

. So

(2.4)

By the logarithmic derivative lemma, we have. Note that and share 1 IM, by Lemma 2.6, so we have

(2.5)

From (2.4) and (2.5) we get (2.3). This proves Lemma 2.7.

Lemma 2.8 [14] Let and be two non-constant meromorphic functions, and be two positive integers. If, then for a constant such that.

By the same reason as in Lemma 5 of [8], we obtain the following lemma.

Lemma 2.9 Let and be two non-constant meromorphic functions. Let be defined as in (1.1)and, and be three  integers with. If, then.

Lemma 2.10 [15] Let and be non-constant meromorphic functions, be two positive integers with, and let be defined as in (1.1), be a small function with respect to

with finitely many zeros and poles. If, and share IM, then

is reduced to a nonzero monomial.

Use the proof of Theorem 3 in [15] and we obtain.

Lemma 2.11 Let and be non-constant meromorphic functions, be two positive integers with

. If, and share IM, then, where and are three constants satisfying.

Lemma 2.12 [16] Let and are relatively prime integers, and let be a complex number such that. Then there exists one and only one common zero of and.

3. Proof of Theorem 1.1

Let, , , , then and share 1 IM and

IM. Suppose that, then, and.

Case 1.. By Lemma 2.4 we have

(3.1)

By Lemma 2.2 with, we obtain

(3.2)

and

(3.3)

Combining (3.1) - (3.3) gives

It follows from Lemma 2.1 and the above inequality that

(3.4)

Similarly we have

(3.5)

Note that .. From (3.4) and (3.5) we deduce that

. (3.6)

Note that and we get (2.3). By Lemma 2.2 with, we obtain

(3.7)

and

(3.8)

From (2.3), (3.7) and (3.8) we get

(3.9)

Combining (3.6) - (3.9) gives

(3.10)

which is a contradiction since. Thus. Similar to the proof of [17, Lemma 3], we obtain 1), or 2).

By Lemma 2.10, the case of 1) is impossible. By Lemma 2.9, we get from 2).

Case 2.. Similar to the proof of Case 1, we get

, (3.11)

which is a contradiction since. Thus. and we have 3), or 4).

For 3), by Lemma 2.11, we get, where and are three constants satisfying.

For 4), By Lemma 2.8, we get for a constant such that. This completes the proof of Theorem 1.1.

4. Proof of Corollaries 1.2 - 1.4

The proof of Corollary 1.2 is the same to the proof of Case 2 of Theorem 1.1, we only need to let. Thus we omit the proof here.

Now we prove Corollary 1.3, Let, similar to (3.10), we get

, (3.12)

which is a contradiction since. Thus and we have 1), or 2).

By Lemma 2.10, the case of (i) is impossible. By Lemma 2.9, we get from 2).

Similar to the proof of Theorem 2 in [14], we get. This proves Corollary 1.3.

Next we prove Corollary 1.4.

According to the proof of Case 1 in Theorem 1.1, we have 1), or 2).

By Lemma 2.10, the case of 1) is impossible. By Lemma 2.9, we get from 2).

Let. If is not a constant, then substitute into and we get

where are distinct roots of the algebraic equation, are distinct roots of the algebraic equation.

Suppose that, then, , where, are co-prime integers andthus, which implies. By Lemma 2.12, there exists one and only one common zero of and, namely. Therefore, there exists at least of different from. Suppose that are different from, then all zeros of have order of at least m. Applying the second fundamental theorem to gives

Note that and we get a contradiction. Thus is a constant. From (4.2) we have and, thus for some constant such that, where. This proves Corollary 1.4.

5. Open Problem

For further study, we pose the following. Problem: What form of implies for some constant?

Acknowledgements

The author would like to thank the referee for his valuable suggestions.

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