Uniqueness of Meromorphic Functions of Differential Polynomials Sharing Two Values IM ()
of the meromorphic function
and by
any quantity satisfying
as ![](https://www.scirp.org/html/1-5300649x\34e652af-97f1-452f-96d5-12948d0079f3.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\7e6ea298-4084-48b8-90ab-c17daefdb6a1.jpg)
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” ![](https://www.scirp.org/html/1-5300649x\b977c40c-df14-4832-9d36-bdfaa1941181.jpg)
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, ![](https://www.scirp.org/html/1-5300649x\f5be2e6c-4c94-44a3-8079-caa7375dc730.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\3e2080af-507c-4f25-b749-49ea20ba35c5.jpg)
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
![](https://www.scirp.org/html/1-5300649x\e81bc595-3913-464d-8f33-f24a73d3e0b6.jpg)
Lemma 2.2 [7] Let
be a non-constant meromorphic function,
be two positive integers. Then
![](https://www.scirp.org/html/1-5300649x\a21d1177-3556-41ad-a7f0-d2fcffce2465.jpg)
![](https://www.scirp.org/html/1-5300649x\4dd32ced-f1d8-49ad-86ce-d94e7e16cc4a.jpg)
Lemma 2.3 [8-10] Let
be a non-constant meromorphic function, and let
be a positive integer. Suppose that
, then
![](https://www.scirp.org/html/1-5300649x\a578e485-a477-4cc3-93e2-4165d13b9730.jpg)
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
![](https://www.scirp.org/html/1-5300649x\1c0ca462-3724-4823-9f32-e979d889076b.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\42cd94df-f79f-4e4b-9796-be022c311864.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\33f6d5db-d7e2-45ab-bdef-1f0195eb5a61.jpg)
with finitely many zeros and poles. If
,
and
share
IM, then ![](https://www.scirp.org/html/1-5300649x\8c92d3fd-828f-482c-822c-cc85dcd01ede.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\e89da1ae-8391-4e70-a168-5c1a5b68e7a4.jpg)
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
![](https://www.scirp.org/html/1-5300649x\394b09aa-db45-46bb-99d3-90740185980a.jpg)
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
![](https://www.scirp.org/html/1-5300649x\47cb5124-55c0-44d2-ae71-e81cc1eaf533.jpg)
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
![](https://www.scirp.org/html/1-5300649x\2a1f6d04-53bb-4911-ad9a-013a07e9e375.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\34e652af-97f1-452f-96d5-12948d0079f3.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\7e6ea298-4084-48b8-90ab-c17daefdb6a1.jpg)
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” ![](https://www.scirp.org/html/1-5300649x\b977c40c-df14-4832-9d36-bdfaa1941181.jpg)
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, ![](https://www.scirp.org/html/1-5300649x\f5be2e6c-4c94-44a3-8079-caa7375dc730.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\3e2080af-507c-4f25-b749-49ea20ba35c5.jpg)
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 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
![](https://www.scirp.org/html/1-5300649x\e81bc595-3913-464d-8f33-f24a73d3e0b6.jpg)
Lemma 2.2 [7] Let
be a non-constant meromorphic function,
be two positive integers. Then
![](https://www.scirp.org/html/1-5300649x\a21d1177-3556-41ad-a7f0-d2fcffce2465.jpg)
![](https://www.scirp.org/html/1-5300649x\4dd32ced-f1d8-49ad-86ce-d94e7e16cc4a.jpg)
Lemma 2.3 [8-10] Let
be a non-constant meromorphic function, and let
be a positive integer. Suppose that
, then
![](https://www.scirp.org/html/1-5300649x\a578e485-a477-4cc3-93e2-4165d13b9730.jpg)
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
![](https://www.scirp.org/html/1-5300649x\1c0ca462-3724-4823-9f32-e979d889076b.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\42cd94df-f79f-4e4b-9796-be022c311864.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\33f6d5db-d7e2-45ab-bdef-1f0195eb5a61.jpg)
with finitely many zeros and poles. If
,
and
share
IM, then ![](https://www.scirp.org/html/1-5300649x\8c92d3fd-828f-482c-822c-cc85dcd01ede.jpg)
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 ![](https://www.scirp.org/html/1-5300649x\e89da1ae-8391-4e70-a168-5c1a5b68e7a4.jpg)
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
![](https://www.scirp.org/html/1-5300649x\394b09aa-db45-46bb-99d3-90740185980a.jpg)
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
![](https://www.scirp.org/html/1-5300649x\47cb5124-55c0-44d2-ae71-e81cc1eaf533.jpg)
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
![](https://www.scirp.org/html/1-5300649x\2a1f6d04-53bb-4911-ad9a-013a07e9e375.jpg)
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.
[1] R. Nevanliana, “Le Theoreme de Picard-Borel at la Theorie des Fonctions Meromorph,” Gauthier-Villars, Paries, 1929.
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