#### Abstract

The main purpose of this paper is to establish the Milloux inequality of -valued meromorphic function from the complex plane to an infinite dimensional complex Banach space with a Schauder basis. As an application, we study the Borel exceptional values of an -valued meromorphic function and those of its derivatives; results are obtained to extend some related results for meromorphic scalar-valued function of Singh, Gopalakrishna, and Bhoosnurmath.

#### 1. Introduction

In the 1970s, the Nevanlinna theory of meromorphic function is extended to the vector-valued meromorphic function from the complex plane to a finite dimensional space (see Ziegler [1]). After that, some works related to vector-valued meromorphic function in finite dimensional spaces were done by [2–5]. In 2006, C. G. Hu and Q. J. Hu [6] established Nevanlinna’s first and second fundamental theorems for an -valued meromorphic function from the disk , , to infinite-dimensional Banach spaces with a Schauder basis. Xuan and Wu [7] established Nevanlinna’s first and second fundamental theorems for an -valued meromorphic function from a generic domain to and generalized Chuang’s inequality. Motivated by [6, 7], Bhoosnurmath and Pujari [8] studied the -valued Borel exceptional values of meromorphic functions, Wu and Xuan [9, 10] studied the characteristic functions, exceptional values, and deficiency of -valued meromorphic function, and Hu [11] surveyed the advancements of the Nevanlinna theory of -valued meromorphic functions and studied its related Paley problems. In this paper, we will generalize Milloux’s inequality (see [12] or [13]) to -valued meromorphic function.

#### 2. The Nevanlinna Theory in Banach Spaces

In this section, we introduce some fundamental definitions and notations of -valued meromorphic function which was introduced by C. G. Hu and Q. J. Hu [6]. See also [7–10].

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 where are the component functions of . Let be an -dimensional projective space of with a basis . The projective operator is a realization of associated with the 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 mapping is called holomorphic (meromorphic) if all component functions of are holomorphic (some of component functions of are meromorphic). The th derivative of is defined by where . A point is called a pole (or point) of if is a pole (or point) of at least one of the component functions of . A point is called a zero of if is a common zero of all the component functions of . A point is called a pole or an -point of of multiplicity which means that in such a point at least one of the meromorphic component functions of has a pole of this multiplicity in the ordinary sense of function theory. A point is called a zero of of multiplicity which means that in such a point all component functions of vanish, each with at least this multiplicity.

An -valued meromorphic function in is said to be of compact projection, if for any given , as sufficiently large in any fixed compact subset .

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 -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 denote the number of -points of in , ignoring multiplicities. Similarly, we can define the counting functions , , and of , , and .

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 denote the number of distinct -points of of order in . 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 We call the -valued meromorphic function admissible if

*Definition 1. *Let be an admissible -valued meromorphic function in . One denotes 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.

In 2006, C. G. Hu and Q. J. Hu [6] proved the following theorems.

Theorem A (the -valued Nevanlinna’s first fundamental theorem). *Let be a nonconstant -valued meromorphic function in , . Then, for , , and ,
**
Here is a function such that
**
and is the coefficient of the first term in the Laurent series at the point .*

Theorem B (the -valued Nevanlinna’s second fundamental theorem). *Let be an admissible -valued meromorphic function of compact projection in , , and be distinct points. Then, for ,
**
where .*

#### 3. Milloux Inequality of -Valued Meromorphic Function

In this section, we will establish the Milloux inequality of -valued meromorphic function and prove the following theorems.

Theorem 2 (Milloux inequality). *Suppose that is an admissible -valued meromorphic function of compact projection in , . Let be distinct points and . Then, for ,
*

In order to prove Theorem 2, we will prove the following general form of Milloux inequality of -valued meromorphic function when the multiple values are considered.

Theorem 3 (general form of Milloux inequality). *Suppose that is an admissible -valued meromorphic function of compact projection in , . Let be distinct points such that and let , and be any positive integers. Then
*

By letting and tend to infinity in (13), we can get Theorem 2. In order to prove Theorem 3, we need the following lemma.

Lemma 4 (see [10]). *Let be of compact projection in ; then, for a positive integer , one has
*

We are now in the position to prove Theorem 3.

*Proof. *We set
then
By [6], we have
From (16) and (17), we can get
Hence, we can get from the above inequality and Lemma 4 that
It follows from Theorem A that
Thus from (19) and (20) we deduce
By Theorem A, we have
Now it follows from Theorems A and B and Lemma 4 that
It follows from (22) and (23) that
A zero of of order is a zero of of order and a zero of of order is a zero of of order . Moreover, zeros of of order are zeros of and so are not zeros of since . Hence
Substituting (25) to (24), we obtain
since
Similarly, we can get
By Lemma 4, we can get
Substituting (27)–(30) into (26), we obtain
Since , , , and are positive integers, it follows from (31) that
Hence, (13) follows from (32).

#### 4. -Valued Borel Exceptional Values of Meromorphic Function and Its Derivatives

Most recently, Bhoosnurmath and Pujari [8] studied the -valued Borel exceptional values of meromorphic functions and gave the following definition.

*Definition 5. *Let be an -valued meromorphic function and is a positive integer. One defines
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 .

Suppose that is an -valued meromorphic function with finite order in . Xuan and Wu [7] proved that the order of is . Hence for any positive integer the order of is . Therefore, we call a vector-valued evB for for distinct zeros of order , if

In this section, we will prove the following theorem.

Theorem 6. *Let be an admissible -valued meromorphic function of compact projection in and the order of is . Suppose that is an -valued evB for for distinct zeros of order , are -valued evB for for distinct zeros of order , and are -valued evB for for distinct zeros of order , where and all of are positive integers. Then
*

*Proof. *By Theorem 3, we obtain
Since is an -valued evB for for distinct zeros of order , is an -valued evB for for distinct zeros of order and is an -valued evB for for distinct zeros of order . Thus there is a such that for any we have
It follows from and (36) and (37) that

Letting in Theorem 6, we can get the following corollary.

Corollary 7. *Let be an admissible -valued meromorphic function of compact projection in and the order of is . Suppose that is an -valued evB for for distinct zeros of order , where is an integer . If there exist , , , such that is an -valued evB for for distinct zeros of order and is a an -valued evB for for distinct zeros of order , where are positive integers, then
*

If , are -valued evB for for distinct zeros, that is, letting tend to infinity in (39), we can get . This means that, for each integer , ,, for all . Hence, we can get the following corollary.

Corollary 8. *Let be an admissible -valued meromorphic function of compact projection in and the order of is . Suppose that , are -valued evB for for distinct zeros. Then, for all positive integers and , for all .*

The corresponding results of Corollaries 7 and 8 for the meromorphic scalar value function were obtained by Gopalakrishna and Bhoosnurmath [14] and Singh and Gopalakrishna [15]. The corresponding results of Corollaries 7 and 8 for the meromorphic scalar value function on annuli were obtained by Chen and Wu [16].

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This research was partly supported by the National Natural Science Foundation of China (Grant no. 11201395). Zuxing Xuan was partly supported by Beijing Natural Science Foundation (Grant No. 1132013).