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The Scientific World Journal
Volume 2013 (2013), Article ID 937584, 7 pages
http://dx.doi.org/10.1155/2013/937584
Research Article

Exceptional Values of Meromorphic Function on Annulus

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China

Received 5 June 2013; Accepted 9 July 2013

Academic Editors: A. Agouzal, A. Barbagallo, and A. Fiorenza

Copyright © 2013 Hong-Yan Xu. 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 study the exceptional values of meromorphic function and its derivative on annulus. We also give some theorems and corollaries about exceptional values of meromorphic function on the annulus, which are the improvement of the previous results given by Chen and Wu.

1. Introduction

It is a very interesting problem on the exceptional values of meromorphic functions in the value distribution theory and argument distributed theory such as the Picard exceptional value and the Borel exceptional value. It is well known that every class of exceptional value is always responding to a singular direction, such as, the Picard exceptional value relating with Julia’s direction and the Borel exceptional value relating with Borel’s direction. In the 1970s, Gopalakrishna and Bhoosnurmath [1, 2] and Singh and Gopalakrishna [3] investigated Borel exceptional values of meromorphic function and its derivative on the whole complex plane. In 2011, Peng and Sun [4] gave some examples on T exceptional value which is an exceptional value relating with T direction.

In fact, Peng, Sun, Singh, Gopalakrishna, and so forth only studied exceptional values of meromorphic functions in the whole complex plane—single-connected region. For meromorphic function on the double-connected domain and several-connected region, there were few papers about its exceptional values. In 2005, Khrystiyanyn and Kondratyuk [5, 6] proposed the Nevanlinna theory for meromorphic functions on annuli (see also [7]). In 2010, Fernández [8] further investigated the value distribution of meromorphic functions on annuli. In 2012, Xu and Xuan [9] studied the uniqueness of meromorphic functions sharing some values on annuli. At the same year, Chen and Wu [10] firstly studied the Borel exceptional values of meromorphic function and its derivative on annulus. In this paper, we will further investigate the exceptional value of meromorphic function and its derivative on annulus and obtain a series of results which are improvement of previous theorems given by Chen and Wu [10].

The structure of this paper is as follows. In Section 2, we introduce the basic notations and fundamental theorems of meromorphic functions on annulus. Sections 3 and 4 are devoted to study the Borel exceptional values of meromorphic function on annulus and give some consequences of our theorems. Section 5 are devoted to study the Borel exceptional values of meromorphic function and its derivative on annulus.

2. Basic Notions in the Nevanlinna Theory on Annuli

Let be a meromorphic function on the annulus , where . The notations of the Nevanlinna theory on annuli will be introduced as follows. Let where and are the counting functions of poles of the function in and , respectively. The Nevanlinna characteristic of on the annulus is defined by Similarly, for , we have in which each zero of the function is counted only once. In addition, we use (or ) to denote the counting function of poles of the function with multiplicities (or ) in , each point counted only once. Similarly, we have the notations , , , , and , .

For a nonconstant meromorphic function on the annulus , where , the following properties will be used in this paper (see [5]): (i), (ii), (iii), for every fixed .

Khrystiyanyn and Kondratyuk [6] also obtained the lemma on the logarithmic derivative on the annulus .

Theorem 1 (see [6] lemma on the logarithmic derivative). Let be a nonconstant meromorphic function on the annulus , where , and let . Then(i) in the case , for except for the set such that ,(ii)if , then for except for the set such that .

In 2005, The second fundamental theorem on the annulus was first obtained by Khrystiyanyn and Kondratyuk [6]. Cao et al. [11] introduce other forms of the second fundamental theorem on annuli as follows.

Theorem 2 ([11, Theorem 2.3] The second fundamental theorem). Let be a nonconstant meromorphic function on the annulus , where . Let be distinct complex numbers in the extended complex plane . Let be positive integers, and let . Then where (i) in the case , for except for the set such that ,(ii) if , then for except for the set such that .

Definition 3. Let be a nonconstant meromorphic function on the annulus , where . The function is called a transcendental or admissible meromorphic function on the annulus which provided that or respectively.

Thus for a transcendental or admissible meromorphic function on the annulus , holds for all except for the set or the set mentioned in Theorem 1, respectively.

Definition 4 (see [10], Definition 2.1). Let be a nonconstant meromorphic function on the annulus , where . Then the order of is defined by

Definition 5 (see [10], Definition 3.1). Let be a nonconstant meromorphic function of order on the annulus , where , and let . Then we say that is(i) an evB (exceptional value in the sense of Borel) for on for distinct zeros of order if ;(ii)an evB (exceptional value in the sense of Borel) for on for distinct zeros if ;(iii)an evB (the Borel exceptional value) for on if , where

In particular, we say that is an evB for on for simple zeros if and is an evB for on for simple and double zeros if .

3. The Main Results

Now, the main theorems of this paper are listed as follows.

Theorem 6. Let be transcendental meromorphic function of order on the annulus , where . If there exist such that are evBs for on for distinct zeros of order , , where , and are positive integers or infinity. Then

Remark 7. Under the assumptions of Theorem 6, if , and , , , since , then we can get This shows that Theorem 6 is an improvement of results given by Chen and Wu [10].

Definition 8. For positive integers , we define that where is the counting function of -points of on where an -point of multiplicity is counted times if and times if . In particular, if , then

Theorem 9. Let be transcendental meromorphic function of order on the annulus , where . If there exist and two positive integers and such that then there exist at most elements which are evBs for on for distinct zeros of order not exceeding .

3.1. Proof of Theorem 6

Proof. For any positive integer or and , we have where and if . Then, from (18) and Theorem 2, we have From Definition 5 and the assumptions of Theorem 6, there exists a constant such that for sufficiently large , Hence, from (19) and (20) and for sufficiently large , we have Thus, for sufficiently large and taking arbitrary , we can get from (21) and the definition of that Since , it follows that for except for the set such that . If , we have where is a constant. Thus, holds for all . If , we can choose an arbitrary satisfying . Thus, from (25) and for sufficiently large , we can get a contradiction to (13) easily.
Therefore, we can get the conclusion of Theorem 6.

3.2. The Proof of Theorem 9

Lemma 10. Let be transcendental meromorphic function of order on the annulus , where . Then

Proof. Suppose that are distinct complex constants. From the definitions of , we have And since , then we have It follows from the definitions of that From (29) and is arbitrary, the proof of Lemma 10 is completed easily.

Proof of Theorem 9. Without loss of generality, we assume that . Next, we will employ reductio ad absurdum to prove the conclusion of Theorem 9. Suppose that there exist elements which are evBs for on for distinct zeros of multiplicity . We know that if is a zero of on of multiplicity for , then is a zero of on of multiplicity . It follows that Therefore, by Theorem 2 we have From the definitions of order and evB of on , there exists a constant such that for sufficiently large , we have From (31), (32), and Lemma 10, for sufficiently large , it follows that for except for the set such that . Since , similar to the argument as in Theorem 6, from (33) we find that holds for all . Since , from (34) and for sufficiently large , we can get which is contradiction with the assumption of Theorem 9.
Thus, this completes the proof of Theorem 9.

4. Some Consequences of Theorems 6 and 9

In this section, we will give some consequences of Theorems 6 and 9. Before we give these results, some definitions will be introduced below.

Definition 11. Let be meromorphic function on the annulus , where . For , then we say that(i) is called an exceptional value in the sense of Nevanlinna (evN for short) for on , if ;(ii) is called a normal value in the sense of Nevanlinna (nvN for short) for on , if .
In addition, similar to the Picard exceptional value in the whole complex plane, we define that is called an exceptional value in the sense of Picard (evP for short) for on , if has at most a finite number of -points on .

Consequence 1. Under the assumptions of Theorem 6, if , from Theorem 6, we get Since and for , it follows that(i)if has an evB for simple zeros which is also an evN for on , then has at most three evBs for simple zeros on ;(ii)if and are two evPs for on then no other element is an evB for for simple zeros on ;(iii) there exist at most four elements which are evBs for simple zeros on since . Moreover, all these four values are nvNs for on .

Consequence 2. Under the assumptions of Theorem 6, let , then(a) if , we have Since and , it follows from (37) that . Thus, if is an evB for on for simple zeros, that is, , then there exist at most two other elements which are evBs for on for distinct simple zeros and double zeros. Furthermore,(i)if , then two other elements evBs are also evNs for on ;(ii)if any of the two other elements and , say , satisfies , then , are also evNs for on ;(b)if , we have From the above inequality and for , we get that , and , , . Thus, if has an evB for simple zeros on , then there exist at most two other elements which are evBs for distinct zeros of multiplicity on . Moreover, all these exceptional values are nvNs for on .

Consequence 3. Under the assumptions of Theorem 6, if , then we have Since , then from (39) we have Thus, it follows that and , . Hence, we have that has at most three evBs for distinct simple and double zeros on . Moreover, all three evBs for distinct simple and double zeros are nvNs for on .

Consequence 4. Under the assumptions of Theorem 6, let .(a) If , then we have Thus, it follows that . So, if there exists an evB for on for distinct and double zeros, say , then there exist at most two other evBs for on for simple zeros, say , . Furthermore, if , , it follows that Thus, we can see that any one of , may not be an evP for on . Furthermore, if is an evP for on . Then , are nvNs for on ;(b) if , , then we have Thus, we can get that . That is, if have two evBs on , say and and is for distinct simple and double zeros, for distinct zeros of order , then has at most one evB for simple zeros on . Furthermore, if , then

It follows that and . Thus, if any equality of these three inequalities holds, then the other two are nvNs for on .

Now, we will give some consequences of Theorem 9 as follows.

Consequence 5. Under the assumptions of Theorem 9, if and we have the following:(i) if and , then there exists at most one element which is an evB for on for simple zeros; in particular, this holds if there exists an satisfying ;(ii)if and , then there exists at most two elements which are evBs for on for simple zeros; in particular, this holds if there exists an satisfying ;(iii)if and , then there exists at most three elements which are evBs for on for simple zeros; in particular, this holds if there exists an satisfying .

Remark 12. Under the assumptions of Theorem 9, from Consequence 5, we have that if there exist four distinct elements which are evBs for on for simple zeros, then and for and .

Consequence 6. Under the assumptions of Theorem 9, if and we have the following:(i) if and , then there exists at most one element which is an evB for on for distinct simple and double zeros; in particular, this holds if there exists an satisfying ;(ii)if and , then there exists at most two elements which are evBs for on for distinct simple and double zeros; in particular, this holds if there exists an satisfying .

Remark 13. Under the assumptions of Theorem 9, from Consequence 6, we have that if there exist four distinct elements which are evBs for on for distinct simple and double zeros, then and for and .

5. Exceptional Value of Meromorphic Function and Its Derivative

In this section, we will study the exceptional value of meromorphic function on and its differential polynomial of the form where , , and is a complex constant.

Theorem 14. Let be transcendental meromorphic function of order on the annulus , where . If 0 and are evBs for on for distinct zeros, then has no evB for simple zeros in .

Proof. From (21), set . Since is a transcendental meromorphic function on , similar to the method of [12, Theorem 1.21], we can get that and have the same order on . By applying Theorem 1 for , for we have Since and 0 and are evBs for on for distinct zeros, then there exists a constant such that for sufficiently large , we have
Suppose that is an evB for on for simple zeros, since is of order . Then for sufficiently large , there exists a constant such that Thus, from (48)–(50), it follows that where and for except for the set such that . Since is of order , then there exists a sequence tending to such that where is a constant. Thus, we can get a contradiction by using the same argument as in Theorem 6.
Therefore, has no evB for simple zeros in .

Acknowledgments

The author was supported by the NNSF of China (no. 61202313) and the Natural Science Foundation of Jiang-Xi Province in China (2010GQS0119 and 20132BAB211001).

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