Some Uniqueness Results of Q-Shift Difference Polynomials Involving Sharing Functions ()
*Supported by the National Natural Science Foundation of China (No.11371139).
1. Introduction
In recent years, many Scholars have been interested in value distribution of difference operators of meromorphic functions (see [1] - [6] ). Furthermore, a large number of papers have studied and obtained the uniqueness results of difference polynomials of meromorphic functions, their shifts and difference operators (see [7] - [12] ). Our purpose in the paper is to study the value distribution for q-shift polynomials of transcendental meromorphic with zero order, and some results about entire functions.
For a meromorphic function
, we always assume that
is meromorphic in the complex plane
. We use standard notations of the Nevanlinna Value Distribution Theory (see [13] ), such as
,
,
,
,
, and define
as the counting function of zero of
, such
that simple zero is counted once and multiple zeros are counted twice. We denote any quantity by
, if it satisfies
, as
outside of a possible exceptional set of r with finite logarithmic measure. In addition, the notation
is the order of growth of
. Let meromorphic function
be a common small function of
and
, suppose that
and
have the same zeros counting multiplicities (ignoring multiplicities), then we say that
and
share
CM(IM).
In this paper, we define a q-shift difference product of meromorphic function
as follows.
(1)
(2)
where
are distinct constants,
be non-zero finite complex constants, let
be a non-zero polynomial, where
are small functions of
. Let
are positive integers and
.
Recently, Liu et al. [14] have considered and proved the uniqueness of q-shift difference polynomials of meromorphic functions.
Theorem A. Let
and
be two transcendental meromorphic functions with
. Let
and
be two non-zero finite complex constants. If
and
share 1 CM, then either
or
, where
satisfying
.
Theorem B. Let
and
be two transcendental meromorphic functions with
. Let
and
be two non-zero finite complex constants. If
and
share 1 IM, then either
or
, where
satisfying
.
First, we will prove the following theorems on value sharing results of q-shift difference polynomials extend the Theorem A, B, as follows:
Theorem 1.1. Let
and
be two transcendental meromorphic functions with
, and let
be a common small function of
and
. If
and
share
CM, then
, where
satisfying
.
Theorem 1.2. Let
and
be two transcendental meromorphic functions with
, and let
be a common small function of
and
. If
and
share
IM, then
, where
satisfying
.
Liu et al. [14] also considered some properties of q-shift difference poly- nomials of entire functions, as follow:
Theorem C. Let
and
be two transcendental entire functions with
, and let
and
are two non-zero finite complex constants, and let
be a non-zero polynomial, where
, are constants, and let m be the number of the distinct zero of
. If
and
share 1 CM, then only one of the following two cases holds:
a)
, where
, and
is greatest common divisor of
, satisfying
. When
, then
, otherwise
.
.
b)
and
satisfy a algebraic equation
, where
(3)
Next, it is easy to derive that
in Theorem C can be replaced by
, as follows
Theorem 1.3. Let
and
be two transcendental entire functions with
, and let
be a common small function of
and
, and let
be the number of distinct zeros of
. If
and
share
CM, then only one of the following results holds:
a)
for a constant
such that
, where
and
is greatest common divisor of
,
,
.
b)
and
satisfy a algebraic equation
, where
(4)
2. Some Lemmas
Lemma 2.1. (see [15] ) Let
be a positive integer, and let
be a transcendental meromorphic function, and let
be small meromorphic functions of
. If
(5)
then
(6)
Lemma 2.2. (see [9] ) Let
and
be two non-zero finite complex numbers, and let
be a nonconstant meromorphic function with
, then
(7)
on a set of logarithmic density 1.
Lemma 2.3. (see [12] ) Let
and
be two non-constant meromorphic functions. Let
and
share 1 IM and
(8)
If
, then
(9)
Lemma 2.4. (see [16] ) Let
and
be two non-constant meromorphic functions. If
and
share 1 CM, then only one of the following results holds:
(10)
Lemma 2.5. (see [14] ) Let
and
be two non-zero finite complex constants, and let
be a non-constant meromorphic function with
, then
(11)
on a set of logarithmic density 1.
Lemma 2.6. (see [14] ) Let
and
be two non-zero finite complex constants, and let
be a nonconstant meromorphic function of zero order, then
(12)
Lemma 2.7. Let
be a non-constant meromorphic function of zero order, and
be defined as in (2). Then
(13)
Proof. Combining Lemma 2.1 with Lemma 2.5, we obtain
(14)
In addition, by Lemma 2.1 and Lemma 2.5, we also get
(15)
which is equivalent to
(16)
Therefore, we get Lemma 2.7.
Lemma 2.8. Let
be an entire function with
, and
be stated as in (2). Then
(17)
Proof. Using the same method as the Lemma 2.7, we can easily to prove.
3. Proof of Theorem
3.1. Proof of Theorem 1.1
Set
,
, than
and
share 1 CM.
Thus by Nevanlinna second fundamental theory, Lemma 2.5 and Lemma 2.7, we have
(18)
Then
(19)
Similarly,
(20)
It follows that
.
Then by Lemma 2.4, we consider three subcases.
Case 1. Suppose that
holds.
Through simple calculation, we have
(21)
In the same way,
(22)
Combining Lemma 2.4, Lemma 2.7, Equations ((21) and (22)), we obtain that
(23)
Then
(24)
Which is impossible, since
.
Case 2. Suppose that
holds, we obtain
(25)
We assume that
. If
(constant), then
, and by substituting
into (25), we obtain that
(26)
Since
is a transcendental meromorphic function, than
. It follows that
.
Suppose that
(constant), then using (25), we deduce that
,
So
(27)
We get a contradiction, since
.
Case 3. Suppose that
holds, then
We define
, we easily get
is non-constant, hence
(28)
We get a contradiction, since
. This implies that
is a constant, which is impossible.
3.2. Proof of Theorem 1.2
Set
,
, So
and
share 1 IM.
Using the same arguments as in Theorem 1.1, we prove that (18)-(22) holds.
We can easily get
(29)
Let
(30)
If
, combining Lemma 2.3, (21), (22) with (29), we obtain
(31)
Then,
(32)
that is impossible, since
. Hence, we get
.
By integrating L twice, we obtain that
(33)
which yields
. From Lemma 2.8, we deduced that
. Next, we will consider the following three subcases.
Case 1.
and
. Suppose that
, by (33), we get
(34)
Combining the second fundamental theory with Lemma 2.5, Lemma 2.7, (29), and (34), we have
(35)
which is impossible, since
. Therefore,
, so
(36)
Then,
. Similarly, we have
(37)
Which is impossible, since
.
Case 2. If
and
, then
obviously. From the proof of case 2 in theorem 1.1, we get
, where
. Therefore, we consider
and
. Then from (33), we obtain
(38)
Using the same discuss as Case 1, we get contradiction.
Case 3. If
and
, then
obviously. Thus from the proof of case 3 in theorem 1.1, we get a contradiction. Therefore, we consider
and
. From (33), we get
(39)
Which is impossible, using the similar method as Case 1.
3.3. Proof of Theorem 1.3
We use the similar method as [14] . By the theorem condition that
and
share 0 CM, hence there exist an entire function
, than
(40)
Since
, than
is a constant.
Rewriting (40)
(41)
If
, we can use Nevanlinnas two fundamental theorems, Lemma 2.5 and Lemma 2.8 to get a contradiction, since
.
So we get
. Rewriting (40)
(42)
Set
, suppose that
(constant), then
. Then we take
into (42) and get
(43)
where
is a non-zero complex constant. And
, since
is transcendental meromorphic function. So
, where
is greatest common divisor of
,
(
).
Suppose that
(constant), Equation (43) imply that
and
satisfy a algebraic equation
, where
(44)
4. Conclusion
In this paper, we obtain some important results about the uniqueness of specific q-shift difference polynomials of meromorphic functions by Nevanlinna and value distribution theory and extend previous results. In addition, we also investigate the problem of value distribution on q-shift difference polynomials of entire functions.
Acknowledgements
Sincere thanks to the members of Xuexue Qian and Yasheng YE for their professional performance, and special thanks to managing editor for a rare attitude of high quality.