Research Article | Open Access

# A Fundamental Inequality of Algebroidal Function

**Academic Editor:**Ahmed El-Sayed

#### Abstract

By using a new mapping of Ahlfors covering surfaces, a fundamental inequality in the angular domain for the algebroidal function is obtained.

#### 1. Introduction and Main Results

In the field of valued distribution, the fundamental inequality is an important tool. For example, it can be used to investigate the singular direction [1]. Using geometric theory, Tsuji firstly obtained the second fundamental theory in an angular domain and proved the existence of Borel direction [2]. The value distribution theory of meromorphic functions was extended to algebroidal functions last century [3]. In 1983, Lv and Gu proved an inequality of algebroidal function for an angular domain [4]. By the inequality, some results of singular direction are obtained; see [5, 6]. In [7], the authors obtained a more accurate inequality for angular domain. In this paper, we will use a new method to simplify and extent an inequality of Tsuji to algebroidal functions.

First, we recall some definitions from [3].

Suppose that are analytic functions with no common zeros in the complex plane. is a bivariate complex function and satisfies For all in the complex plane, the equation has complex roots . Then, (1.1) defines a -valued algebroidal function ; see [3, 8]. If , then is called -valued integral algebroidal function. If is irreducible, correspondingly is called -valued irreducible algebroidal function (note that is a meromorphic function, if ). Now we suppose that is an irreducible algebriodal function defined by (1.1).

If , and the -degree equation and its partial derivative have no common roots (i.e., is not a multiple root of ), then is said to be a regular point. The set of all regular points is called the regular set, denoted by . Its complementary set : is called the critical set. Obviously, includes all branch points of (see [3]).

The domain of a -valued irreducible algebriodal function is a connected Riemann surface [8], and its single-valued domain is denoted by . A point in is and sets lying over and are and . Let and be the number of zeros, counted according to their multiplicities, of in and , respectively. Let be the number of distinct zeros in , and let be the number of branch points in . Similarly, we can define and . Let

Similarly, we can define , and . From [3], we know that and .

In this paper, we will prove the main theorem.

Theorem 1.1. *Let be a -valued algebroidal function in region . are different complex numbers on the sphere with radius . For , and , we have
*

By the inequality in Theorem 1.2, we will immediately have the following.

Theorem 1.2. *For a meromorphic function (a 1-valued algebroidal function with no branch points) defined by (1.1) satisfying
**
it has at least one Nevanlinna direction, that is, there exists , such that holds for any finitely many deficient value , where
*

#### 2. Some Lemmas

First, it is easy to prove the following.

Lemma 2.1. *Suppose that , then there exists , such that
**
where
*

Lemma 2.2. *Suppose that and , then *(1)*the mapping maps the unit disc to the square , where is constant; *(2)* maps , where , into a symmetrical convex region in ; *(3)* in Lemma 2.2 satisfies
*

*Proof. *Obviously, (1) holds. By the definition of , we have .

In order to compute , first we prove that maps , , , to four symmetry axes of Q, where . Let ( is fixed), ,

Hence, when , , , , , respectively, see Figure 1.

Then, we compute . Since is the only intersection point of the lines and . The center of the square , , is that of the curves and . Then, conforms onto .

Hence,

(2) At last we prove that is a convex region, see Figure 1.

For a fixed , by (2.4),

Set . When , , , by Lemma 2.1,
where .

Hence,

Therefore, the image of is a descending convex curve. By the symmetry of square, is a smooth curve, and is a convex symmetric figure in the square .

Then, we obtain Lemma 2.2.

Lemma 2.3. *Suppose that mappings
**
Then, the mapping maps the region into the square , and , where , , , and
*

*Proof. *The conclusion is equivalent to . We prove the lemma by two cases.*Case I*. When
then
is a convex symmetric figure if there exists a , such that
then Lemma 2.3 holds, see Figure 2.

In fact, for any , ,
where
where

By Lemma 2.1, we have
where

Let

Combing (3.1) (*note *that by (3.1), we have ),

Therefore, a vertex of . By Lemma 2.2, .*Case II.* When
since is a convex symmetric figure, we also have Lemma 2.3, see Figure 3.

For the convenience of readers, we prove the following lemma again, it can be found in [9].

Lemma 2.4. * (1) Let , then
**(2) Put , , then
*

*Proof. *(1) Since and are holomorphic functions, then

For
then
Hence,

Thus,

(2) By
we have
Similarly,

Then,

Therefore,

Lemma 2.5 (see [10]). *Let be a connected covering surface on , is bounded by different points with radius , then
**
where is the length of and is Euler characteristic of , is the area of and
*

Lemma 2.6 (see [10]). * Let be a sphere with radius , be bounded by different points with radius and then
**
where , .*

*Proof. *Suppose that . Then
where

Hence, by the Jacobian determinant, we have

By , we have Lemma 2.6.

#### 3. Proof of Theorem 1.1

* Proof. *Set . It conforms the unit disc to the sector and the interior of to , where , , and
Hence conforms to the sphere .

Put . Then by M. Hurwite Formula, we have
Put and . Then
By Lemma 2.5, it follows that
Now we will prove
For any , and , we have
where , , .

By (3.6), for any , we have
Therefore
Put , . Hence

By
where
we obtain

(1) If
then by (3.4)
that is,
Hence,
Therefore

(2) If there is a , such that
then
Equation (3.17) holds.

By (3.17) and Lemma 2.3, we have

#### 4. Proof of Theorem 1.2

* Proof. *By the hypothesis of Theorem 1.2, there exists an increasing sequence (, when ), such that

Then, there exist some , such that for arbitrary ,
holds. We claim that is the Nevanlinna direction.

Otherwise, for a positive number , there exist some , such that
By the definition of , we have
There exists , such that for any ,

Hence, for defined earlier, when is sufficiently large,
By Theorem 1.1, we have
Hence,
Hence,
which contradicts (4.2). Therefore, Theorem 1.2 holds.

#### Acknowledgments

The research is supported by the National Natural Science Foundation of China (no. 11101096), Guangdong Natural Science Foundation (no. S2012010010376), and the Startup Foundation for Doctors of Guangdong University of Technology (no. 083063).

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#### Copyright

Copyright © 2012 Yingying Huo and Daochun Sun. 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.