Journal of Applied Mathematics and Physics
Vol.06 No.11(2018), Article ID:88554,9 pages
10.4236/jamp.2018.611188
Uniqueness of Meromorphic Functions Whose Differential Polynomials Share One Value
Jin Tao, Xinli Wang
College of Science, University of Shanghai for Science and Technology, Shanghai, China
Copyright © 2018 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: September 29, 2018; Accepted: November 16, 2018; Published: November 19, 2018
ABSTRACT
In this paper, we prove a uniqueness theorem of meromorphic functions whose some nonlinear differential shares 1 IM with powers of the meromorphic functions, where the degrees of the powers are equal to those of the nonlinear differential polynomials. This result improves the corresponding one given by Zhang and Yang, and other authors.
Keywords:
Differential Polynomials, Meromorphic Functions, Uniqueness Theorems
1. Introduction
The meromorphic function mentioned in this paper refers to the meromorphic function over the entire complex plane. Let f and g be two non-constant meromorphic functions. means Linear measure finite set. means . CM is the abbreviation of common multiplicities. And IM is the abbreviation of ignored multiplicities. These concepts can be found in the literature [1] . Let a be a finite complex number, if and have the same zero point and the same number of weights, then f and g CM share a. If and have the same zero point without counting the number, then f and g IM share a [2] . In addition, the following definitions are required: let p be a positive integer, and . Next
means f has a weight less than p count function of the weight of the value point a within . means corresponding reduced count function; means the weight of f is not less than p count function of the weight of the value point a within . means corresponding reduced count function. Suppose k is a non-negative number. Mark defined as follows. See the literature [3] for details.
Before, Xiaomin Li and Zhitao Wen expanded Jilong Zhang’s theorem, where changes to , so when , that is Zhang’s theorem. Similarly, in this paper, we continuously change to , which contained . So we expended Xiaomin Li and Zhitao Wen’s theorem.
In 2008, Lianzhong Yang and Jilong Zhang proved the following theorems:
Theorem A [4] Suppose f is a non-constant entire function, is a positive integer, if and CM share 1, then .
Theorem B [4] Suppose f is a non-constant meromorphic function, is a positive integer, if and CM share 1, then .
Recently Zhang Jilong improved the above theorem. Get the following result:
Theorem C [5] Suppose f is a non-constant entire function, is a positive integer, if and CM share 1, then .
Theorem D [5] Suppose f is a non-constant meromorphic function, is a positive integer, if and CM share 1, then .
Li Xiaomin and Wen Zhitao have improved on the basis of Zhang Jilong’s theorem, as follows.
Theorem E [5] Suppose f is a non-constant meromorphic function, k is a positive integer, n is a positive integer and satisfies , if and CM share 1, then .
Theorem F [5] Suppose f is a non-constant meromorphic function, k is a positive integer, n is a positive integer and satisfies , if and IM share 1, then .
Now we mainly improve the theorem of Li Xiaomin. Which that changes and to and . We get the following theorem:
Theorem 1 Suppose f is a non-constant meromorphic function, k is a positive integer, n is a positive integer and satisfies , if and IM share 1, and the zeros of with multiplicity 2 at least., then .
2. Some Lemmas
Lemma 1 [6] Suppose F and G are non-constant meromorphic functions, let
and suppose , if F and G IM share 1, then
Lemma 2 [7] Let f be a non-constant meromorphic function, and
where are constants, then .
Lemma 3 Let f be a non-constant meromorphic function, and are two positive integers. Let and . If F and G IM share 1, then .
Proof According to Lemma 2, we obtain
(1)
It can be seen from the above formula,
(2)
due to , we have
(3)
According to the second basic theorem and (2)
The above formula is combined with (1) to get
(4)
According to (3) and (4), we have
(5)
According to (2) and (5), we can get the conclusion of Lemma 3.
Lemma 4 [8] Let f be a non-constant meromorphic function, are two positive integers. The zero point of is at least 2, then
Lemma 5 Let f be a non-constant meromorphic function, , p are two positive integers. The zero point of is at least 2. Let and , if F and G IM share 1, then
a)
b) ;
c) .
Proof According to Lemma 4, we have
This leads to the conclusion (a), obtained from the definition of the and Lemma 3:
This leads to conclusions (b), the same reason
Combine and the q form in Lemma 5, we can get (c).
Lemma 6 Suppose F and G are non-constant meromorphic functions, and satisfy and . If F and G IM share a non-zero constant a, then or .
Proof
Suppose , . (6)
Let H be defined by Lemma 1. The following two discussions,
Case 1 Suppose , then , let
(7)
If ,
(8)
where is a constant, if
By (6) and (8), we get .
So , contradictory with the assumption of case 1.
Therefore, .
So , by (8), we get
(9)
(10)
According to the second basic theorem and (6) (8) (9) (10) we get
(11)
By lemma 2, we have
(12)
Then
(13)
If , then , contradiction.
If V was not always equal to 0, (7) can be rewritten into
(14)
Suppose is a pole of f with multiplicity p, then is pole of F with multiplicity . and is zero of with multiplicity at least. is zero of with multiplicity at least.
So is a zero of V with multiplicity at least.
Then
(15)
The following two sub-cases are discussed:
Sub-case 1.1 suppose
(16)
If , we have
where is a constant. Then
. (17)
Suppose , , , contradiction;
Suppose , , , contradiction.
So , . (18)
If , ,
So .
Then
Obviously impossible.
Suppose U is not always equal to 0, let be a zero of f with multiplicity q, then is a zero F with multiplicity nq and is zero of with multiplicity at least. is zero of with multiplicity at least.
So is a zero of U with multiplicity at least.
So
(19)
Also
Then
(20)
If , and , one of the two forms is established. Then , substituting the above formula is obviously impossible.
Or , contradiction.
So , we get
, then
(21)
Case 2
Situation 2.1
If , we get
,
According to Lemma 6, we get or .
Firstly, if , , , conclusion established.
Secondly, if , , .
Obviously f is entire function.
And
, contradiction.
Situation 2.2
If , we get .
So .
If , we get , then .
And if ,
We get , that contradict with .
Suppose ,
If we get .
If , we get , then .
If , we get
And
We get , which contradicts with .
Therefore, Theorem 1 is proved.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Tao, J. and Wang, X.L. (2018) Uniqueness of Meromorphic Functions Whose Differential Polynomials Share One Value. Journal of Applied Mathematics and Physics, 6, 2264-2272. https://doi.org/10.4236/jamp.2018.611188
References
- 1. Hayman, W.K. (1964) Meromorphic Functions. Claredon Press, Oxford.
- 2. Yi, H.X. and Yang, C.J. (1995) Theorem on the Uniqueness of Meromorphic Functions. Science Press, Beijing.
- 3. Alzahary, T.C. and Yi, H.X. (2004) Weighted Sharing Three Values and Uniqueness of Meromorphic Functions. Journal of Mathematical Analysis and Applications, 295, 247-257. https://doi.org/10.1016/j.jmaa.2004.03.040
- 4. Yang, L.Z. and Zhang, J.L. (2008) Non-Existence of Meromorphic Solusions of Fermat Type Functional Equation. Aequationes Mathematicae, 76, 140-150. https://doi.org/10.1007/s00010-007-2913-7
- 5. Yang, L.Z. and Zhang, J.L. (2009) A Power of a Meromorphic Function Sharing a Small Function Sharing a Small Function with Its Derivative. Annales Academiae Scientiarum Fennicae Mathematica, 34, 249-260.
- 6. Yi, H.X. (1997) Uniqueness Theorems for Meromorphic Functions Whose n-th Derivatives Share the Same 1-Points. Complex Variables, Theory and Application, 34, 421-436. https://doi.org/10.1080/17476939708815064
- 7. Yang, C.C. (1972) On Deficiencies of Differential Polynomial. Mathematische Zeitschrift, 125, 107-112. https://doi.org/10.1007/BF01110921
- 8. Lahiri, J. and Sarkar, A. (2004) Uniqueness of a Meromorphic Function and Its Derivative. Journal of Inequalities in Pure and Applied Mathematics, 5, 20-21.