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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 426302, 11 pages
http://dx.doi.org/10.1155/2012/426302
Research Article

The Filling Discs Dealing with Multiple Values of an Algebroid Function in the Unit Disc

1Beijing 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
2Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received 7 August 2011; Accepted 18 December 2011

Academic Editor: Gabriel Turinici

Copyright © 2012 Zu-Xing Xuan and Nan Wu. 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

In this paper, by using the potential theory we prove the existence of filling discs dealing with multiple values of an algebroid function of finite order defined in the unit disc.

1. Introduction and Main Result

The value distribution theory of meromorphic functions due to Hayman (see [1] for standard references) was extended to the corresponding theory of algebroid functions by Selbreg [2], Ullrich [3], and Valiron [4] around 1930. The filling discs of an algebroid function are an important part of the value distribution theory. For an algebroid function defined on -plane, the existence of its filling discs was proved by Sun [5] in 1995. In 1997, for the algebroid functions of infinite order and zero order, Gao [6] obtained the the corresponding results. The existence of the sequence of filling discs of algebroid functions dealing with multiple values, of finite or infinite order, was first proved by Gao [7, 8]. The existence of filling discs in the strong Borel direction of algebroid function with finite order was proved by Huo and Kong in [9]. Compared with the case of , it is interesting to investigate the algebroid functions defined in the unit disc, and there are some essential differences between these two cases. Recently, the first author [10] has investigated this problem and confirmed the existence of filling discs for this case. In this note, we will continue the work of Xuan [10] by considering the case dealing with multiple values and get more precise results.

Let be the -valued algebroid function defined by irreducible equation where are entire functions without any common zeros. The single-valued domain of definition of is a -valued covering of the -plane, a Riemann surface, denoted by . A point in , whose projection in the -plane is , is denoted by . The part of , which covers the disc , is denoted by .

Denoteis called the mean covering number of into -sphere under the mapping . And is conformal invariant. Let be the number of zeros of , counted according to their multiplicities in denotes the number of zeros with multiplicity of in , each zero being counted only once.

Let where is the boundary of . The characteristic function of is defined by In view of [4], we have The order of algebroid function is defined by

In this paper we assume that is the -sphere, and is a constant which can stand for different constant. Let be the number of the branch points of in , counted with the order of branch. Write

Valiron is the first one to introduce the concept of a proximate order for a meromorphic function with finite positive order and is called type function of or such that is nondecreasing, piecewise continuous, and differentiable, and

For an algebroid function of finite positive order, we can apply the same method to get its type function .

Our main result is the following.

Theorem 1.1. Suppose that is the -valued algebroid function of finite order in defined by (1.1) and (≥ ) is an integer, then there exists a sequence of discs where Such that for each except for those complex numbers contained in the union of spherical discs each with radius , where .
The discs with the above property are called filling discs dealing with multiple values.

Remark 1.2. In [10], the result says that . Theorem 1.1 is really the improvement of [10].

Remark 1.3. The existence of filling discs in Borel radius of meromorphic functions was proved by Kong [11]. In view of our theorem, we can get the similar results of [11] (when ). But we must point out that the structure and definition of filling discs between this paper and [11] are different. There are also some papers relevant to the singular points of algebroid functions in the unit disc (see [1214]).

2. Two Lemmas

Lemma 2.1 (see [15] or [16]). Suppose that is the -valued algebroid function in defined by (1.1), is an integer and are distinct points given arbitrarily in -sphere, and the spherical distance of any two points is no smaller than , then for any , one has

Combining the potential theory with Lemma 2.1, one proves Lemma 2.2, which is crucial to the theorem.

Lemma 2.2. Suppose that is the -valued algebroid function of finite order satisfying in defined by (1.1) and is an integer. For any , there exists , such that for any , put where stands for the inter part of .
Then, among , there exists at least one pair , such that , and in , except for those complex numbers contained in the union of spherical discs each with radius .

Proof. Suppose the conclusion is false. Then there exists a sequence , where . For any , any and , there exist complex numbers which satisfy that the spherical distance of any two of those points is no smaller than . Denote For any mentioned above, we have
For any , let , then we have .
For any given positive integers and , set Then Thus there exists which are related to . We can assume , such that Set Then we have Since covers and twice at most. We obtain Obviously, each can be mapped conformally to the unit disc such that the center of is mapped to , and the image of is contained in the disc (<1). Since all are similar, is independent of . Since is conformally invariant, in view of Lemma 2.1, we obtain
For sufficiently large integer . Thus we get where is a constant.
For any integer , we have . We can choose one as such that . For a certain sufficiently fixed large inter , we have This yields the following: where .
Hence Next, we deduce the following: when .
In view of for , we have Dividing both sides of (2.13) by and integrating it from to , we have Note that is fixed, we see that is a finite constant.
Hence, Then In view of [3], we know that We obtain where is a constant.
Dividing both sides of the above inequality by , we have We note that In view of the properties of the , we obtain Letting in (2.24), we have that is, Letting , and , we obtain This is contradictory to , and the lemma is proved.

3. Proof of the Theorem

Proof. Choose .
In view of Lemma 2.2, there exists , and Let Then Set Take then Put Then In view of Lemma 2.2, for each , except for those complex numbers contained in the union of spherical discs each with radius . Theorem 1.1 is proved.

Remark 3.1. By using the same method, we can prove the existence of filling discs for -quasimeromorphic mappings whose general case is carefully discussed in another paper.

Acknowledgments

The authors would like to thank the referee for his/her many helpful suggestions on an early version of the paper. Z.-X. Zuxing is supported in part by Beijing Municipal Research Foundation for The Excellent Scholars Ministry (2011D005022000009); Science and Technology Research Program of Beijng Municipal Commission of Education (KM201211417011); Funding Project for Academic Human Resources Development in Beijing Union University.

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