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Abstract and Applied Analysis
Volume 2011, Article ID 518972, 13 pages
http://dx.doi.org/10.1155/2011/518972
Research Article

An Inequality of Meromorphic Vector Functions and Its Application

1School of Mathematics and Statistics, Xianning University, Xianning 437100, China
2School of Mathematics and Computer Science, Xinyu University, Xinyu 338004, China

Received 17 July 2011; Accepted 7 November 2011

Academic Editor: Gerd Teschke

Copyright © 2011 Wu Zhaojun and Chen Yuxian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Firstly, an inequality for vector-valued meromorphic functions is established which extend a corresponding inequality of Milloux for meromorphic scalar-valued function (1946). As an application, the relationship between the characteristic function of a vector-valued meromorphic function and its derivative is studied, results are obtained to extend some related results for meromorphic scalar-valued function of Weitsman (1969) and Singh and Gopalakrishna (1971).

1. Introduction of Vector-Valued Meromorphic Function

In 1980s, Ziegler [1] established Nevanlinna’s theory for the vector-valued meromorphic function in finite dimensional spaces. After Ziegler some works related to vector-valued meromorphic function were done in 1990s [24]. In this section, we shall introduce the following fundamental notations and results of vector-valued Nevanlinna theory which were quoted from Ziegler [1].

We denote by the usual dimensional complex Euclidean space with the coordinates , the Hermitian scalar product and the distance

Let be complex valued functions of the complex variable , which are meromorphic and not all constant in the Gaussian plane , or in a finite disc Thus in (we put ), a vector-valued meromorphic function is given, which does not reduce to the constant zero vector . The th derivative of are defined by

For such a function, the notations “zero,” “pole,” and “multiplicity” are defined as in the scalar case of only one meromorphic function . More explicitly, in the punctured vicinity of each point , the vector function can developed into a Laurent series where the coefficients are vectors

In order to introduce the Nevanlinna theory of vector-valued meromorphic function, we will denote by "" the ideal element of the Aleksandrov one-point compactification of (the two real infinities will be denoted by and , resp.). Now, if in the above Laurent expansion, then will be called a pole or an -point of of multiplicity ; in such a point at least one of the meromorphic component functions has a pole of this multiplicity in the ordinary sense of function theory, so that in itself is not defined. If in Laurent expansion, then is called a zero of of multiplicity ; in such a point , all component functions vanish, each with at least this multiplicity.

Let or denote the number of poles of in and denote the number of -points of in , counting with multiplicities. Define the volume function associated with vector-valued meromorphic function , and the counting function of finite or infinite -points by respectively. Next, we define Let or denote the number of poles of in and denote the number of -points of in , ignoring multiplicities. Define the counting function of finite or infinite -points by respectively.

If is a vector-valued meromorphic function in the whole complex plane, then the order and the lower order of are defined by

We call the vector-valued meromorphic function admissible if

Definition 1.1. For a meromorphic function (vector-valued or scalar-valued), we denote by any quantity such that without restriction if is of finite order and otherwise except possibly for a set of values of of finite linear measure.

Definition 1.1 quoted from [2]. In [1], Ziegler established the following first main theorem, logarithmic derivative lemma, and deficient values theorem for meromorphic vector function.

Theorem A. Let be a meromorphic vector function in . Then for , , then

Theorem B. Let be a nonconstant meromorphic vector function in . Then

By the second main theorem, Ziegler [1] studies the following deficiency theorem for meromorphic vector function. For any vector , we define the number by putting and by putting

Theorem C. Let be an admissible meromorphic vector function in . Then the set is at most countable and summing over all such points we have

2. A Fundamental Inequality of Meromorphic Vector Function

For meromorphic scalar-valued function , Milloux [5] has proved the following theorem.

Theorem D. If is a nonconstant meromorphic scalar-valued function in Gaussian complex plane and if , are distinct elements of (where is any positive integer), then

For some alternative proofs of Theorem D, see [6] or [7]. It is natural to consider whether there exists a similar results if meromorphic scalar-valued function is replaced by meromorphic vector-valued function . In this section, the main contribution is to extend Theorem D to vector-valued meromorphic function by referring the method of [1, 7].

Theorem 2.1. Let be an admissible meromorphic vector function in and if , are distinct elements of (where is any positive integer), then

Proof. Put We can get Put Let for the moment be fixed. Then we get in every point where the inequality for . Therefore, the set of points on which is determined by (2.6) is either empty or any two such sets for different have empty intersection. In any case, Because of it follows that so that by (2.4) Thus by Theorem B, we have It follows from Theorem A that Thus from (2.12) and (2.13), we deduce Adding to both sides, where is formed with the zeros of which are not zeros of any of . Since , we have Since it follows that

3. Characteristic Function of Derivative of Meromorphic Vector Function

Let be a meromorphic scalar-valued function in . The characteristic function of derivative of with has been studied by Edrei [8], Shan and Singh [9], Singh and Gopalakrishna [7], Singh and Kulkarni [10] and Weitsman [11]. For example, Edrei [8] and Weitsman [11] have proved the following theorem.

Theorem E. Let be a transcendental meromorphic scalar-valued function of finite order and assume and . Then

If is replaced by , Singh and Gopalakrishna [7] and Singh and Kulkarni [10] have proved the following theorem.

Theorem F. Let be a transcendental meromorphic scalar-valued function of finite order and assume . Then for every .

It is natural to consider whether there exists a similar results if meromorphic scalar-valued function is replaced by meromorphic vector-valued function . In this section, the main purpose is to extend the above theorems to vector-valued meromorphic function by referring the method of [1, 7].

Theorem 3.1. Let be an admissible meromorphic vector function of finite order in and assume . Then for every .

Proof. Now, basic estimates in vector-valued Nevanlinna theory [1] or [4] yields By Theorem B and the above inequality, we have
Let be a sequence of distinct vector in containing all the vector of . From Theorem 2.1, for any positive integer , we have Hence Thus Since was arbitrary, we have This and (3.5) yield
Let and an infinite sequence of distinct elements of which includes every satisfying and . Then
Let be any integer . From Generalized Second Main Theorem (see [1], Page 126), we have Hence Thus Since this holds for all , letting and combining (3.11), we get So
For every , Let be any integer . From Generalized Second Main Theorem (see [1], Page 126), we have Hence Thus Since this holds for all , letting and combining (3.11), we get So

From Theorem 3.1, we have the following corollary

Corollary 3.2. Let be an admissible meromorphic vector function of finite order in and assume . Then

Corollary 3.3. Let be an admissible meromorphic vector function of finite order in and assume and . Then

Corollary 3.4. Let be an admissible meromorphic vector function of finite order in and assume . Then for every .

Proof. Since for every and Theorem C, it follows that, if , then and for every . Hence follows by Theorem 3.1.
Now, for every , Further Hence
Similarly, Further Hence

From Corollary 3.4, we have the following corollary.

Corollary 3.5. Let be an admissible meromorphic vector function of finite order in and assume . Then

Corollary 3.6. Let be an admissible meromorphic vector function of finite order in and assume and . Then

Acknowledgments

This research was partially supported by the NSF of Jiangxi Province (Grant 2010GZC0187), by NSF of Educational Department of the Hubei Province (Grant T201009, Q20112807), and by Grant PY1002 of Xianning University.

References

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