Applied Mathematics
Vol.08 No.08(2017), Article ID:78579,11 pages
10.4236/am.2017.88084
Some Uniqueness Results of Q-Shift Difference Polynomials Involving Sharing Functions*
Xuexue Qian, Yasheng Ye
Department of Mathematics, College of Sciences, University of Shanghai for Science and Technology, Shanghai, China
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: July 26, 2017; Accepted: August 18, 2017; Published: August 21, 2017
ABSTRACT
In this paper, we mainly study the uniqueness of specific q-shift difference polynomials and of meromorphic functions, which share a common small function and get the corresponding results. In addition, we also investigate the problem of value distribution on q-shift difference polynomials of entire functions.
Keywords:
Value Distribution, Meromorphic Functions, Difference Polynomials, Uniqueness
*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.
Cite this paper
Qian, X.X. and Ye, Y.S. (2017) Some Uniqueness Results of Q-Shift Difference Polynomials Involving Sharing Functions. Applied Mathematics, 8, 1117-1127. https://doi.org/10.4236/am.2017.88084
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