Milloux Inequality of E-Valued Meromorphic Function

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).