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On Uniqueness of Meromorphic Functions with Multiple Values in Some Angular Domains
Journal of Inequalities and Applications volume 2009, Article number: 208516 (2009)
Abstract
This article deals with problems of the uniqueness of transcendental meromorphic function with shared values in some angular domains dealing with the multiple values which improve a result of J. Zheng.
1. Introduction
A transcendental meromorphic function is meromorphic in the complex plane and not rational. We assume that the readers are familiar with the Nevanlinna theory of meromorphic functions and the standard notations such as Nevanlinna deficiency
of
with respect to
and Nevanlinna characteristic
of
. And the lower order
and the order
are in turn defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ1_HTML.gif)
For the references, please see [1]. An is called an IM (ignoring multiplicities) shared value in
of two meromorphic functions
and
if in
,
if and only if
. It is Nevanlinna [2] who proved the first uniqueness theorem, called the Five Value Theorem, which says that two meromorphic functions
and
are identical if they have five distinct IM shared values in
. After his very fundamental work, the uniqueness of meromorphic functions with shared values in the whole complex plane attracted many investigations (see [3]). Recently, Zheng in [4] suggested for the first time the investigation of uniqueness of a function meromorphic in a precise subset of
, and this is an interesting topic.
Given pair of real numbers
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ2_HTML.gif)
we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ3_HTML.gif)
Zheng in [4] proved the following theorem.
Theorem A
Let and
be both transcendental meromorphic functions, and let
be of finite order
and such that for some
and an integer
. For
pair of real numbers
satisfying (1.2) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ4_HTML.gif)
where , assume that
and
have five distinct IM shared values in
. If
, then
However, it was not discussed whether there are similar results dealing with multiple values in some angular domains. In this paper we investigate this problem.
We use to denote the set of zeros of
in
, with multiplicities no greater than
, in which each zero counted only once.
Our main result is what follows.
Theorem 1.1.
Let and
be both transcendental meromorphic functions, and let
be of finite order
and such that for some
and an integer
. For
pair of real numbers
satisfying (1.2) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ5_HTML.gif)
where , assume that
are
distinct complex numbers, and let
be positive integers or
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ6_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ8_HTML.gif)
where . If
then
2. Proof of Theorem 1.1
First we introduce several lemmas which are crucial in our proofs. The following result was proved in [5] (also see [6]).
Lemma 2.1 (see [5]).
Let be transcendental and meromorphic in
with the lower order
and the order
. Then for arbitrary positive number
satisfying
and a set
with finite linear measure, there exists a sequence of positive numbers
such that
(1),
,
(2)
(3).
A sequence satisfying (1), (2), and (3) in Lemma 2.1 is called Polya peak of order
outside
in this article. For
and
define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ10_HTML.gif)
The following result is a special version of the main result of Baernstein [7].
Lemma 2.2.
Let be transcendental and meromorphic in
with the finite lower order
and the order
and for some
. Then for arbitrary Polya peak
of order
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ11_HTML.gif)
Although Lemma 2.2 was proved in [7] for the Polya peak of order , the same argument of Baernstein [7] can derive Lemma 2.2 for the Polya peak of order
.
Nevanlinna theory on angular domain will play a key role in the proof of theorems. Let be a meromorphic function on the angular domain
, where
. Nevanlinna defined the following notations (see [8]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ12_HTML.gif)
where and
are the poles of
on
appearing according to their multiplicities.
is called the angular counting function of the poles of
on
and Nevanlinna's angular characteristic is defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ13_HTML.gif)
Throughout, we denote by a quantity satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ14_HTML.gif)
where denotes a set of positive real numbers with finite linear measure. It is not necessarily the same for every occurrence in the context [9].
Lemma 2.3.
Let be meromorphic on
. Then for arbitrary complex number
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ15_HTML.gif)
and for an integer ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ16_HTML.gif)
and .
Lemma 2.4.
Let be meromorphic on
. Then for arbitrary
distinct
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ17_HTML.gif)
where the term will be replaced by
when some
.
We use to denote the zeros of
in
whose multiplicities are no greater than
and are counted only once. Likewise, we use
to denote the zeros of
in
whose multiplicities are greater than
and are counted only once.
Lemma 2.5.
Let be meromorphic on
and let
be
positive integers. Then for arbitrary
distinct
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ18_HTML.gif)
where the term will be replaced by
when some
.
Proof.
According to our notations, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ19_HTML.gif)
By Lemma 2.4,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ20_HTML.gif)
and (i) follows.
Furthermore, , and on combining this with (i), we get (ii).
Proof of Theorem 1.1.
Suppose . For convenience, below we omit the subscript of all the notations, such as
and
. By applying Lemma 2.5 to
and (1.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ21_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ22_HTML.gif)
This implies that . We have also (2.14) for alternation of
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ23_HTML.gif)
By (1.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ24_HTML.gif)
We assume that . By the same argument we can show Theorem 1.1 for the case when
. By applying Lemma 2.3 and (2.16), we estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ25_HTML.gif)
The following method comes from [10]. But we quote it in detail here because of its independent significance. Note that . We need to treat two cases.
(I)?? Then
. And by the inequality (1.5), we can take a real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ26_HTML.gif)
where , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ27_HTML.gif)
Applying Lemma 2.1 to gives the existence of the Polya peak
of order
of
such that
, and then from Lemma 2.2 for sufficiently large
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ28_HTML.gif)
since . We can assume for all the
, (13) holds. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ29_HTML.gif)
Then from (2.18) and (2.20) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ30_HTML.gif)
It is easy to see that there exists a such that for infinitely many
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ31_HTML.gif)
We can assume for all the , (2.23) holds. Set
. Thus from the definition (2.1) of
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ32_HTML.gif)
On the other hand, by the definition (2.4) of and (2.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ33_HTML.gif)
Combining (2.24) with (2.25) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ34_HTML.gif)
Thus from (1.5) in Lemma 2.1 for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ35_HTML.gif)
This is impossible.
(II)?? Then
By the same argument as in (I) with all the
replaced by
, we can derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ36_HTML.gif)
This is impossible. Theorem 1.1 follows.
Remark 2.6.
In Theorem A, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ37_HTML.gif)
so Theorem A is a special case of Theorem 1.1. Meanwhile, Zheng in [4, pages 153–154] gave some examples to indicate that the conditions are necessary. So the conditions in theorem are also necessary.
Corollary 2.7.
In Theorem 1.1,
(i)if , then
,
(ii)if then
,
(iii)if , then
,
(iv)if , then
,
(v)if , then
,
(vi)if , then
,
Corollary 2.8.
Let and
be both transcendental meromorphic functions and let
be of finite lower order
and such that for some
and an integer
. For
pair of real numbers
satisfying (1.2) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ38_HTML.gif)
where , assume that
are
distinct complex numbers satisfying that
, where
is an integer or
. If
, then
.
Corollary 2.9.
Let and
be both transcendental meromorphic functions and let
be of finite lower order
and such that for some
and an integer
. For
pair of real numbers
satisfying (1.2) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F208516/MediaObjects/13660_2009_Article_1918_Equ39_HTML.gif)
where , assume that
are
distinct complex numbers satisfying that
,
, then
.
Question 1.
For two meromorphic functions defined in , there are many uniqueness theorems when they share small functions (
is called a small function of
if
) (see [3]). So we ask an interesting question: are there similar results when they share small functions in some precise domain
?
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Acknowlegment
The work is supported by NSF of China (no. 10871108).
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Xuan, ZX. On Uniqueness of Meromorphic Functions with Multiple Values in Some Angular Domains. J Inequal Appl 2009, 208516 (2009). https://doi.org/10.1155/2009/208516
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DOI: https://doi.org/10.1155/2009/208516