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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 260506, 15 pages
http://dx.doi.org/10.1155/2012/260506
Research Article

Characteristic Functions and Borel Exceptional Values of -Valued Meromorphic Functions

1School of Mathematics and Statistics, Hubei University of Science and Technology, Hubei, Xianning 437100, China
2Beijing Key Laboratory of Information Service Engineering, Department of General Education, Beijing Union University, No. 97 Bei Si Huan Dong Road, Chaoyang District, Beijing 100101, China

Received 27 April 2012; Accepted 16 September 2012

Academic Editor: Michiel Bertsch

Copyright © 2012 Zhaojun Wu and Zuxing Xuan. 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

The main purpose of this paper is to investigate the characteristic functions and Borel exceptional values of -valued meromorphic functions from the to an infinite-dimensional complex Banach space with a Schauder basis. Results obtained extend the relative results by Xuan, Wu and Yang, Bhoosnurmath, and Pujari.

1. Introduction and Preliminaries

In 1980s, Ziegler [1] succeeded in extending the Nevanlinna theory of meromorphic functions to the vector-valued meromorphic functions in finite dimensional spaces. Later, Hu and Yang [2] established the Nevanlinna theory of meromorphic mappings with the range in an infinite-dimensional Hilbert spaces. In 2006, C.-G. Hu and Q. Hu [3] established the Nevanlinna's first and second main theorems of meromorphic mappings with the range in an infinite-dimensional Banach spaces with a Schauder basis. Recently, Xuan and Wu [4] established the Nevanlinna's first and second main theorems for an -valued meromorphic mapping from a generic domain to an infinite-dimensional Banach spaces with a Schauder basis.

In [4], Xuan and Wu also proved Chuang's inequality (see, e.g., [5]) of -valued meromorphic mapping in the whole complex plane, which compares the relationship between and , and also obtained that the order and the lower order of -valued meromorphic mapping and those of its derivative are the same. In Section 2, we shall prove that Chuang's inequality is valid for -valued meromorphic mapping in the unit disc and prove that for any infinite-order -valued meromorphic function defined in the unit disc has the same Xiong's proximate order as its derivative .

In [5], Yang obtained much stronger results than those of Gopalakrishna and Bhoosnurmath [6] for the Borel exceptional values of meromorphic functions dealing with multiple values. In Section 3, we shall extend Le Yang's result to -valued meromorphic functions of finite and infinite orders in

In the following, we introduce the definitions, notations, and results of [3, 4] which will be used in this paper.

Let be an infinite dimension complex Banach space with Schauder basis and the norm . Thus, an -valued meromorphic function defined in , can be written as Let be an -dimensional projective space of with a basis . The projective operator is a realization of associated with basis.

The elements of are called vectors and are usually denoted by letters from the alphabet: . The symbol denotes the zero vector of . We denote vector infinity, complex number infinity, and the norm infinity by , and , respectively. A vector-valued mappings is called holomorphic (meromorphic) if all are holomorphic (some of are meromorphic). The th derivative of is defined by A point is called a “pole" (or point) of if is a pole (or point) of at least one of the component functions . A point is called a “zero" of if is a zero of all the component functions . A point is called a pole or an -point of of multiplicity , meaning that 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. A point is called a zero of of multiplicity , meaning that in such a point all component functions vanish, each with at least this multiplicity.

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

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

If is an -valued meromorphic function in , then the order and the lower order of are defined by

Lemma 1.1. Let be a positive and continuous function in which satisfies . Then there exists a continuously differentiable function , which satisfies the following conditions.(i) is continuous and nondecreasing for and tends to as . (ii)The function satisfies the following: (iii).

Lemma 1.1 is due to K. L. Hiong (also Qinglai Xiong) and is called the proximate order of Hiong. A simple proof of the existence of was given by Chuang [7]. Suppose that is an -valued meromorphic function of infinite order in the unit disk . Let and . From (ii) and (iii) in Lemma 1.1, we have Here, the functions and are called the proximate order and type function of , respectively.

Definition 1.2. An -valued meromorphic function in is of compact projection, if for any given , has sufficiently larg in any fixed compact subset .
Throughout this paper, we say that is an -valued meromorphic function meaning that is of compact projection. C.-G. Hu and Q. Hu [3] established the following Nevanlinna's first and second main theorems of -valued meromorphic functions.

Theorem 1.3. Let be a nonconstant -valued meromorphic function in . Then for , Here, is a function satisfying that and is the coefficient of the first term in the Laurent series at the point .

Theorem 1.4. Let be a nonconstant -valued meromorphic function in and be distinct points. Then for ,

If , then holds as without exception if has finite order and otherwise as outside a set of exceptional intervals of finite measure . If the order of is infinite and is the proximate order of , then holds as without exception.

If , then holds as without exception if has finite order and otherwise as outside a set of exceptional intervals of finite measure .

In all cases, the exceptional set is independent of the choice of .

2. Characteristic Function of -Valued Meromorphic Functions in the Unit Disc

In [4], Xuan and Wu proved the following.

Theorem A. Let be a nonconstant -valued meromorphic function and . Then for and , one has where is a positive constant.

Theorem B. Let be a nonconstant -valued meromorphic function. Then we have

Theorem C. For a nonconstant -valued meromorphic function of order , one has .

In this section, we shall prove that Theorems A, B, and C are valid for -valued meromorphic function in the unit disc .

Lemma 2.1. Let be an -valued meromorphic function defined in the unit disc, and . If , then there exists a , such that for any , one has

Lemma 2.2. Let be an -valued meromorphic function defined in the unit disc, and let . Then there exists a positive number , such that for , one has

Lemmas 2.1 and 2.2 are due to Xuan and Wu [4] for the -valued meromorphic function defined in the whole complex plane. From the proof of Xuan and Wu [4], we know that Lemmas 2.1 and 2.2 are also valid for the -valued meromorphic function defined in the unit disc .

Lemma 2.3. Let be a nonconstant -valued meromorphic function and . Suppose that , then when sufficiently tends to 1, one has

Proof.

Lemma 2.4 (see [4]). Let be a nonconstant -valued meromorphic function and , and a curve from the origin along the segment to , and along turn a rotation to . Then for any , one has where .

Lemma 2.5 (see [3]). Let be a nonconstant -valued meromorphic function in . Then for , where is a sufficiently large constant.

We are now in the position to establish the main results of this section.

Theorem 2.6. Let be a nonconstant -valued meromorphic function and . Then for and any real function , when sufficiently tend to 1, one has

Proof. Denote , we can get Applying Lemma 2.1 to and combining Lemma 2.3, we can find a real number such that for any , one has In view of Lemma 2.2, there is a such that for any , one has From the origin along the segment to and along , turn a rotation to . We denote this curve by . In virtue of Lemma 2.4, we have holds for any , where . In virtue of (2.11), (2.12), and (2.13), we have Hence,

Theorem 2.7. Let be a nonconstant -valued meromorphic function and . Then for any , one has

Proof. By Lemma 2.5, we have

Theorem 2.8. For a nonconstant -valued meromorphic function of order , one has .

Theorem 2.8 only discussed the -valued meromorphic function of finite order. In fact, for any -valued meromorphic function of infinite order, we have the following.

Theorem 2.9. If is a nonconstant -valued meromorphic function of order , then the proximate orders of and are the same.

Proof. Let , in view of Theorems 2.6 and 2.7, we can easily derive Theorem 2.9.

3. -Valued Borel Exceptional Values of Meromorphic Functions in

Some definitions in this section can be found in [8].

Definition 3.1. Let be an -valued meromorphic function and , if is a positive integer, let or denote the number of distinct poles of of order in , and let denote the number of distinct -points of of order in . Similarly, we can define the counting functions , , and of , , and .

Definition 3.2. Let be an -valued meromorphic function and . If , we define If , we define

Definition 3.3. Let be an -valued meromorphic function and and is a positive integer, we say that is an (i)-valued evB (exceptional value in the sense of Borel) for for distinct zeros of order if ;(ii)-valued evB for for distinct zeros if ;(iii)-valued evB for (for the whole aggregate of zeros) if .
In [5], Yang proved the following result.

Theorem D. Let be a meromorphic function with finite order and be positive integers. is called a pseudo-Borel exceptional value of of order if If has distinct pseudo-Borel exceptional values of order , then

It is natural to consider whether there exists a similar result, if meromorphic function is replaced by -valued meromorphic function . In this section, we extend the above theorem to -valued meromorphic function in .

Theorem 3.4. Let be an -valued meromorphic function with finite order , any system of distinct elements in , and any system such that is a positive integer or . If is an -valued evB for for distinct zeros of order , then

Proof. By Theorem 1.4, we have holds for . For any , we have Using (3.7) and (7) in (3.6), we get Therefore, we have By hypothesis, we have If , then there is a positive number , such that for , we can get Using (3.11) to (3.9), we have If, then by Theorem 1.4 and (3.12), we can get a contradiction . So If , then there is a positive number , such that for , we can get Using (3.14) to (3.9), we have If, then by Theorem 1.4 and (3.15), we can get a contradiction . So

From the proof of Theorem 3.4, we can get the following.

Corollary 3.5. Let be a nonconstant -valued meromorphic function. Then for any system of distinct elements in and any system such that is a positive integer or , we have the following: (1)if all of in , then (2)if one of is , say . Then,

Remark 3.6. If , let and , and in Theorem 3.4. We can get the following result by Bhoosnurmath and Pujari [8].

Theorem E. Let be an -valued meromorphic function of order . If there exist distinct elements in such that are -valued evB for for distinct zeros of order , are -valued evB for for distinct zeros of order , are -valued evB for for distinct zeros of order , where , , and are positive integers, then

Bhoosnurmath and Pujari [8] pointed out that Theorem E is valid for . In fact, Definition 3.3 is not well in the case of . In the case of is an -valued evB for if and only if is finite. When is infinite, we shall give the following definitions.

Definition 3.7. Let be an -valued meromorphic function of infinite order and is a proximate order of and . We say that is an (i)-valued evB (exceptional value in the sense of Borel) for for distinct zeros of order if (ii)-valued evB for for distinct zeros if (iii)-valued evB for (for the whole aggregate of zeros) if

Theorem 3.8. Let be an -valued meromorphic function of infinite order and is a proximate order of , any system of distinct elements in , and any system such that is a positive integer or . If is an -valued evB for for distinct zeros of order , then

Proof. By Corollary 3.5, we have By hypothesis, there exists a positive number such that Using (3.25) to (3.26), we have If , then by Theorem 1.4 and (3.27), we can get a contradiction. So

Acknowledgments

The first author is supported in part by the Science Foundation of Educational Commission of Hubei Province (Grant nos. T201009, Q20112807). The second author is supported in part by the Science and Technology Research Program of Beijing Municipal Commission of Education (KM201211417011).

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