1. Introduction
The value distribution theory of meromorphic functions occupies one of the central places in Complex Analysis which now has been applied to complex dynanics, complex differential and functional equations, Diophantine equations and others.
In his excellent paper [1], W.K. Hayman studied the value distribution of certain meromorphic functions and their derivatives under various conditions. Among other important results, he proves that if f(z) is a transcendental meromorphic function in the plane, then either f(z) assumes every finite value infinitely often, or every derivative of f(z) assumes every finite nonzero value infinitely often. This result is known as Hayman’s alternative. Thereafter, the value distribution of derivatives of transcendental functions continued to be studied.
In this paper, we study the value distribution of transcendental meromorphic functions, all but finitely many of whose zeros have multiplicity at least
, where
is a positive integer.
In 2008, Liu et al. [2] proved the following results.
Theorem A Let
be an integer, let
be a meromorphic function of infinite order
in
, and let
, where
is a polynomial. Suppose that 1) all zeros of
have multiplicity at least
, except possibly finitely many, and 2) all poles of
are multiple, except possibly finitely many.
Then
has infinitely many zeros.
Theorem B Let
be an integer, let
be a meromorphic function of finite order
in
, and let
, where
is a polynomial. Suppose that 1) all zeros of
have multiplicity at least
, except possibly finitely many, and 2)
.
Then
has infinitely many zeros.
In the present paper, we prove the following result, which is a significant improvement of Theorem 1.
Theorem 1 Let
be an integer, let
be a meromorphic function of order
in
, and let
, where
is a polynomial. Suppose that all zeros of
have multiplicity at least
, except possibly finitely many. Then
has infinitely many zeros.
Theorem 1 and Theorem 2 taken together imply the following result.
Theorem 2 Let
be an integer, let
be a meromorphic function in
, and let
, where
is a polynomial. Suppose that 1) all zeros of
have multiplicity at least
, except possibly finitely many, and 2)
as
.
Then
has infinitely many zeros.
2. Notation and Some Lemmas
We use the following notation. Let
be complex plane and
be a domain in
. For
and
,
and
. We write
in
to indicate that the sequence
converges to
in the spherical metric uniformly on compact subsets of
and
in
if the convergence is in the Euclidean metric.
Let
be a meromorphic function in
. Set
(1.1)
The Ahlfors-Shimizu characteristic is defined by
![](https://www.scirp.org/html/2-5300607x\d05af128-64d2-4a53-bf31-ad676619686c.jpg)
Remark Let
denote the usual Nevanlinna characteristic function. Since
is bounded as a function of
, we can replace
with
in the paper.
The order
of the meromorphic function
is defined as
![](https://www.scirp.org/html/2-5300607x\2cbee6ab-d12d-48a4-a1ed-bf489dc373fe.jpg)
Lemma 1 [3] Let
a sequence of holomorphic functions in
such that
locally uniformly in
, where
is univalent in
. Let
be a sequence of functions meromorphic in
such that for each
1) all zeros of
have multiplicity at least
; and 2)
.
Then
is quasinormal of order 1 in
. If, moreover, no subsequence of
is normal at
, then
![](https://www.scirp.org/html/2-5300607x\99436d1d-eeaf-4c54-ae7c-ab3489b645b3.jpg)
locally uniformly in
and there exists
such that
for all
.
Remark Since Lemma 1 is not stated explicitly in [3], let us indicate how it follows from the results of that paper. The proof that
is quasinormal of order 1 is essentially identical to that of Theorem
of [3]. That proof also shows that condition (b) of Lemma 7 in [3] holds for
. It then follows from Lemma 7 that
locally uniformly on
. The bound on
follows from Lemma 9 of [3]. See also [4, Remark on page 484].
Lemma 2 [5, Lemma 2] Let
be a family of functions meromorphic in
, all of whose zeros have multiplicity at least
, and suppose that there exists
such that
whenever
. Then if
is not normal at
, there exist, for each
1) points
,
;
2) functions
; and 3) positive numbers ![](https://www.scirp.org/html/2-5300607x\99d3f3f8-f6e9-4813-a406-d91c5d4e66a1.jpg)
such that
in
, where
is a nonconstant meromorphic function in
all of whose zeros have multiplicity at least
, such that
.
Lemma 3 Let
be a meromorphic function of order
in
, then there exist
and
such that
![](https://www.scirp.org/html/2-5300607x\f0fc690c-cf44-4351-820f-3839e86e5555.jpg)
Proof We claim that there exist
and
such that
(1.2)
Otherwise there would exist
and
such that
![](https://www.scirp.org/html/2-5300607x\efda05c3-dd34-4ca6-88ca-87019429a4ad.jpg)
for all
. From this follows
![](https://www.scirp.org/html/2-5300607x\41b4352d-035e-4b88-ba7e-bb7fba4af1fe.jpg)
and hence
![](https://www.scirp.org/html/2-5300607x\add7b529-813a-4bcd-8926-8efc723e60dc.jpg)
Now we have
which contradicts the hypothesis that
.
Observing that
hence there exists a sequence
such that
and
as
. Let
. Obviously,
and
, and hence
as
.
Lemma 4 Let
and
. Let
be a transcendental meromorphic function, all of whose zeros have multiplicity at least
. Set
. Suppose that
. Then there exists a sequence
and
such that
![](https://www.scirp.org/html/2-5300607x\06a1730d-b375-4fef-8c9f-23ea523435b3.jpg)
as
.
Proof Since
and
, we have
. By Lemma 3, there exist
and
such that
![](https://www.scirp.org/html/2-5300607x\3ee56845-92e0-4614-99d4-504c4a8d6418.jpg)
Set
. Clearly,
. Thus
is not normal at 0. Obviously, all zeros of
have multiplicity at least
in
, and hence all zeros of
have multiplicity at least
in
for sufficiently large
. Using Lemma 2 for
, there exist points
, and positive numbers
and a subsequence of
(that we continue to call
) such that
![](https://www.scirp.org/html/2-5300607x\11f49d55-fa15-4882-bd7d-1bc8721d2fc4.jpg)
in
, where
is a nonconstant meromorphic function in
, all of whose zeros have multiplicity at least
.
We claim that
, where
is a constant. Otherwise,
, where
and
are constants. Then, either
is a constant function, or all zeros of
have multiplicity at most
. A contradiction.
Let
be not a zero or pole of
, and let
. Now we have
![](https://www.scirp.org/html/2-5300607x\1fabdacc-ad81-435a-8969-2bc14051aaf8.jpg)
where
. Since
and
is not a zero or pole of
, we have
,
and
as
, where
.
Set
and
, where
. Clearly,
![](https://www.scirp.org/html/2-5300607x\ebc8e535-aad4-4816-8d3b-767fce000156.jpg)
where
satisfying
as
.
Now, we have
and
![](https://www.scirp.org/html/2-5300607x\3511a377-dd85-48d8-9ff2-491d9cd681bc.jpg)
Set
. Obviously,
and
, and hence
as
.
3. Proof of Theorem
Proof We assume that
has at most finitely many zeros and derive a contradiction. Let
as
, where
and
.
Set
. By Lemma 4, there exists a sequence
and
such that
(1.3)
and
(1.4)
Set
. By (1.4),
(1.5)
Hence, no subsequence of
is normal at
.
Since
has at most finitely many zeros, we have for sufficiently large
,
![](https://www.scirp.org/html/2-5300607x\f779ff9d-f3ea-41cc-aba2-71e8c7c9adee.jpg)
Observing that
![](https://www.scirp.org/html/2-5300607x\9596de77-20bf-42e8-afa7-7557ee6970b2.jpg)
in
. It follows from Lemma 1 (applied to
in
), and there exists
such that for all ![](https://www.scirp.org/html/2-5300607x\23b447e9-a454-4574-be7f-4a0622d6b82b.jpg)
(1.6)
Set
. Then
![](https://www.scirp.org/html/2-5300607x\293321c3-f4a8-43df-91ea-b36ebfca7358.jpg)
and hence
(1.7)
Using the simple inequality
![](https://www.scirp.org/html/2-5300607x\5d4bef6f-20f4-4d68-8a98-5df08c089aa6.jpg)
for
, we have
(1.8)
The second term on the right of (1.7) is
(1.9)
Putting (1.7), (1.8), and (1.9) together, we have for
and sufficiently large
,
(1.10)
It follows from (1.1), (1.6), and (1.10),
![](https://www.scirp.org/html/2-5300607x\ecf84240-4b41-46b6-8817-1ba7e53d63bd.jpg)
Thus,
![](https://www.scirp.org/html/2-5300607x\04b183bb-f998-45fe-a265-e12cd6cb6c15.jpg)
which contradicts (1.3).
Acknowledgements
This work was supported by National Natural Science Foundation of China (No.11001081, No.11226095).
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