Research Article | Open Access
Zu-Xing Xuan, Nan Wu, "On the Nevanlinna's Theory for Vector-Valued Mappings", Abstract and Applied Analysis, vol. 2010, Article ID 864539, 15 pages, 2010. https://doi.org/10.1155/2010/864539
On the Nevanlinna's Theory for Vector-Valued Mappings
The purpose of this paper is to establish the first and second fundamental theorems for an -valued meromorphic mapping from a generic domain to an infinite dimensional complex Banach space with a Schauder basis. It is a continuation of the work of C. Hu and Q. Hu. For defined in the disk, we will prove Chuang's inequality, which is to compare the relationship between and . Consequently, we obtain that the order and the lower order of and its derivative are the same.
In 1980s, Ziegler  established Nevanlinna's theory for the vector-valued meromorphic functions in finite dimensional spaces. After Ziegler some works in finite dimensional spaces were done in 1990s [2–4]. In 2006, C. Hu and Q. Hu  considered the case of infinite dimensional spaces and they investigated the -valued meromorphic mappings defined in the disk . In this article, by using Green function technique, we will consider this theory defined in generic domain (see Section 2). In Section 3, motivated by the work of [6–8], we will prove Chuang's inequality, which is to compare the relationship between and . Consequently, we obtain that the order and the lower order of and its derivative are the same. This is an extension of an important result for meromorphic functions.
2. First and Second Fundamental Theorem in Generic Domains
Let be a complex Banach space with a Schauder basis and the norm . Thus an -valued meromorphic mapping defined in a domain can be written as The elements of are called vectors and are usually denoted by letters from the alphabet: . The symbol 0 denotes the zero vector of . We denote vector infinity, complex number infinity, and the norm infinity by , , and , respectively. A vector-valued mapping is called holomorphic(meromorphic) if all are holomorphic (meromorphic). The th derivative () and the integration of are defined by respectively. We assume that . A point is called a “pole” or “-point” of if is a pole of at least one of the component functions . We define when is a pole. A point is called “zero” of if all the component functions have zeros at .
Remark 2.1. The integrals are well defined because the set of singularities making meaningless is zero measurable.
In order to make our statement clear, we first recall some knowledge of Green functions.
Definition 2.2. Let be a domain surrounded by finitely many piecewise analytic curves. Then for any , there exists a Green function, denoted by , for with singularity at which is uniquely determined by the following: (1) is harmonic in ;(2)in a neighborhood of , for some function harmonic in ;(3), on the boundary of .
By we denote the boundary of and the inner normal of with respect to . Using Green function we can establish the following general Poisson formula for the -valued meromorphic mapping, which is similar with [5, Lemma ] (see [9, Theorem ], or [10, Theorem ]). We do not give the details here.
Theorem 2.3. Let be an -valued meromorphic mapping, which does not reduce to the constant zero element . Then where , are the zeros of and are the poles of according to their multiplicities.
Remark 2.4. A simple inspection to the -valued case shows that is not harmonic for a holomorphic (or meromorphic) -valued function. Therefore we have an additional term in formula (2.2).
Following Theorem 2.3, we introduce some notations. where is a point in and are the poles of in appearing according to their multiplicities, . Define
is called the Nevanlinna characteristic function of with the center .
Next, we give the first (FFT) and the second (SFT) fundamental theorems for .
Theorem 2.5 (FFT). Let be an -valued meromorphic mapping on . Then for a fixed vector and for any such that , one has where
Proof. We can rewrite Theorem 2.3 as follows: Applying this formula to the function , we can prove the theorem.
Theorem 2.6 (SFT). Let be an -valued meromorphic mapping on , let be distinct vectors, and let . Then, where Furthermore, one has the following form:
According to the property of the logarithm function, we get
Denote , and fix . Then we obtain
So either the set of points on which is determined by (2.14) is empty or any two of some sets for different have intersection. In any case, on we have Since it follows that From (2.12), we get Since is nonconstant vector, does not reduce to the constant zero element . Applying FFT to , we can obtain Using this formula, we have On the other hand, we have The two inequalities above give That is to say, Adding to the above inequality and applying FFT, we can formulate where Since , (2.24) can be written as If contains , (2.26) also holds. Let , and substitute with ; then we have (2.26), where , and
Next we establish Hiong King-Lai's inequality for .
Theorem 2.7. Let be an -valued meromorphic mapping on , , let be three finite vectors, and let , . Then one has
3. The Vector-Valued Mapping and Its Derivative
In this section, we will discuss the value distribution theory of defined in the disk . We will prove Chuang's inequality. According to (2.3), we have the following terms:
where denotes the number of poles of in . The order and the lower order of an -valued meromorphic mapping are defined by The following lemma is well known.
Lemma 3.1 (see [11, Boutroux-Cartan Theorem]). Let be complex numbers. Then the set of the point satisfying can be contained in several disks, denoted by ; the total sum of its radius does not exceed .
The next lemma is a special case of Theorem 2.3.
Lemma 3.2 (see ). Let be an -valued meromorphic mapping, which does not reduce to the constant zero element . Then, for a , one has Here are all the zeros and poles counting their multiplies of in D.
In order to obtain the relationship between and , we should first establish the following two lemmas.
Lemma 3.3. Let be a nonzero -valued meromorphic mapping, and . If and are two positive numbers, and , then there exists a , such that for any one has
Lemma 3.4. Let be a nonzero -valued meromorphic mapping, and let be three positive numbers. Then there exists a positive number , and for , one has
Proof. Let be the poles of in . By Boutroux-Cartan Theorem, we have except for some points contained in a pack of disks whose radius does not exceed . Then there exists a circle such that and . Thus (3.14) holds on . For any , we have
Now we are in the position to establish the following Chuang's inequality.
Theorem 3.5. Let be a nonzero -valued meromorphic mapping and . Then for and , one has where is a positive constant.
Proof. Take a such that and denote . Applying Lemma 3.3 to , we can find a real number such that , and we have
In view of Lemma 3.4, for a fixed we have
From the origin along the segment to , and along turn a rotation to . We denote this curve by , and its length is .
We notice that is continuous on . As in , is an -dimensional projective space of with a basis . The projection operator is a realization of associated to the basis and . We have and . Therefore, since is finite dimensional, there exists (appearing in the inequality , where and are any two norms on ) such that where . Thus, we have
In virtue of [6–8], every meromorphic mapping with values in a Banach space with a Schauder basis and the projections are convergent in its natural topology; that is, they converge uniformly to in any compact subset of ( being the set of poles the in ). Thus for large enough, we have A similar argument to implies that for large enough where .
Combining (3.20), (3.21), and (3.22) and the fact that the compact set , we get Then In virtue of (3.13) and (3.17), we have Therefore, Thus we have
The following result says that we can also control the by .
Theorem 3.6. Let () be a nonconstant -valued meromorphic mapping. Then one has
Corollary 3.7. For a nonconstant -valued meromorphic mapping , One has , .
The authors would like to thank the referee for his/her many helpful comments and suggestions on an early version of the manuscript. The work is supported by NSF of China (Grant no. 10871108). The first author is also supported partially by NSF of China (Grant no. 10926049).
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