Uniqueness of Meromorphic Functions Whose Differential Polynomials Share One Value ()
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.