#### Abstract

By applying Ahlfors theory of covering surface, we establish a fundamental inequality of meromorphic function dealing with multiple values in an angular domain. As an application, we prove the existence of some new singular directions for a meromorphic function , namely a Bloch direction and a pseudo-T direction for .

#### 1. Introduction

In this paper, meromorphic function always means a function meromorphic in the whole complex plane. Given a meromorphic function , the theory of value distribution of developed in the two ways: one is the module distribution and the other is angular distribution. For the module distribution of a meromorphic function, there are three main theorems, that is, the Picard theorem, the Borel theorem, and the Nevanlinna second fundamental theorem. The fundamental concept in the angular distribution is singular direction. Singular direction is a concept of localizing value distribution in onto a sector containing a single ray emanating from the origin say. A Julia direction and a Borel direction are refinements of the Picard theorem and the Borel theorem, respectively. Corresponding to the Nevanlinna second fundamental theorem, a new singular direction, called T direction, was recently introduced in Zheng [1]. When multiple values were considered, Yang [2] proved the following theorems related to the module distribution of meromorphic function. In order to introduce the main results of Yang, we give some notations (see [2]) as the following.

Let denote a nonconstant meromorphic function, an arbitrary complex number, and a positive integer. We use or to denote the zeros of in , whose multiplicities are no greater than , counted according to their multiplicities. Likewise, we use or to denote those zeros in , whose multiplicities are greater than , counted according to their multiplicities. The corresponding counting functions are denoted by or and or . Let be a meromorphic function with order , be an arbitrary number, and be a positive integer. If then is called a pseudo-Borel exceptional value of of order .

In [2], Yang has proved the following theorems.

Theorem A. *Let be a meromorphic function with order and let be positive integers. If has distinct pseudo-Borel exceptional values of order , then
*

Theorem B. *Let be a nonconstant meromorphic, be distinct complex numbers, and be positive integers. Then
**
where is the Nevanlinna error term.*

In this paper, we will research the singular directions corresponding to Theorems A and B.

#### 2. A Theorem on Covering Surface

In this section, we will give a theorem on covering surface. We firstly introduce the following notations (see Tsuji [3]).

In this paper, the Riemann sphere of diameter 1 is denoted by . Let be a finite covering surface of , consisting of a finite number of sheets, and be bounded by a finite number of analytic Jordan curves (some of which may reduce to single points), and the spherical distance between any two circular curves and is . The part of the boundary of , which does not lie above the boundary of , is called the relative boundary of and denote its spherical length by . Let be a domain on , whose boundary consists a finite number of points or analytic closed Jordan curves, and let be the part of , which lies above . We denote the spherical area of , , and by , and , respectively. We put Under the above notation, we have the following Ahlfors covering Theorem.

Lemma 1 (see Tsuji [3]). *For any finite covering surface of , one has
**
where is a constant which depends on only.*

Recently, Sun [4] has proved a precise version of Lemma 1 and proved that , where is a constant.

Lemma 2 (see Sun [5]). *Let be a simply connected finite covering surface of the unite sphere , and let be disjoint spherical disks on , where the spherical distance of any pair of is at least . Let be the number of simply connected islands (see Tsuji [3, Page 252]) in ; then
**
where is the length of the relative boundary of and is a constant.*

Theorem 3. *Let be a simply connected finite covering surface of the unite sphere , and let be positive integers. Let be disjoint spherical disks with radius on and without a pair of such that their spherical distance is less than and let be the number of simply connected islands in , which consisted of no more than sheets; then
**
where is the length of the relative boundary of .*

*Proof. *It is easy to verify that
where is the number of simply connected islands in , which consist of no less than sheets. Hence,
Since the spherical area of is , it follows from Lemma 1 that
Note that and ; we can get
Adding two sides of the above expression from 1 to , we have
Combining Lemma 2 and the above expression, Theorem 3 follows.

#### 3. A Fundamental Inequality of Meromorphic Functions in an Angular Domain

The Ahlfors-Shimizu characteristic is important in this paper. Let us recall its definition. Suppose that is a nonempty subset of ; we denote When , we write by . Then from Theorem 1.4 in [6], we have And the difference is a bounded function of , so that both the characteristic function and are interchangeable. Denote the following angular domain by When is a sector , we denote and For any and , let be the number of zeros, counted according to their multiplicities, of in the sector , and let be the number of zeros with multiplicities , of in the sector , where is any positive integer. Similarly, note the number of poles of by and . Denote In addition, we also need the notations (see [7])

In this section, we will establish a fundamental inequality for meromorphic functions in an angular domain. Firstly, we give the following lemma.

Lemma 4. *Suppose that is a meromorphic function and be positive integers, and are distinct points on and without a pair of such that their spherical distance is less than . be the number of zeros of , which are consisted of not more than multiplicities, then
*

*Proof. *Let be a spherical disk with the center with radius on . By Theorem 3, we have
Note that , whenever in the island of or in the peninsula of . Therefore, Lemma 4 follows.

We are now in the position to establish the main result in this section.

Theorem 5. *Let be a meromorphic function and positive integers. If are distinct points on , then one has
**
for any , , where is a constant depending only on , , and .*

*Proof. *Put and . Using Lemma 4, we have
where , which depends only on , that is, only on , and
Hence
Denote the left expression of (25) by ; thus
We claim the fact that
In fact, it follows from the definition of and Schwarz's inequality that
Noting is an increasing function of , we see that then there exists a , such that , when , and , when . For , by (25) and (27),
that is,
Integrating each side of the inequality leads to
Thus
On the case of , the above inequality is obviously valid because of . Replacing in the above inequality with its explicit expression, we see that (21) is established. Therefore
where .

Lemma 6 (Zhang [7]). *Under the condition of Theorem 5, one has
**
or
**
with at most one exceptional set of , where consists of a series of intervals and satisfies
**
In particular, if the order of is , then
*

From Theorem 3 and Lemma 6, we can write the result in Theorem 3 as If the order of is , then the inequality will be

#### 4. Bloch Direction of Meromorphic Functions

In this section, we will research the singular direction corresponding to Theorem A. Suppose that is a meromorphic function of infinite order. Then, there is a real function called an Hiong's proximate order (see [8]) of , which has the following properties. (i) is continuous and nondecreasing for and tends to as . (ii) The function satisfies the condition

For a meromorphic function of infinite order, Zhuang Qitai (or Chuang Chitai) [9] gives the following definition of Borel direction and Bloch direction.

*Definition 7. *Let be a meromorphic function of infinite order and an order of . A direction is called a Borel direction of order of if, no matter how small the positive number is, for each value , one has
except for at most two exceptional values . A direction is called a Bloch direction of order of if, for any number , any system of distinct values and, any system such that is a positive integer or and that
there exists at least one integer such that

For the connection of Borel direction and Bloch direction of meromorphic function of infinite order, Chuang [9] has proved the following theorem.

Theorem C. *Let be a meromorphic function of infinite order and an order of . Then every Borel direction of order of is a Bloch direction of order of .*

It is natural to consider whether there exists a similar result, if meromorphic function of order infinity is replaced with meromorphic function of order . In this section we extend the above theorem to meromorphic function of order .

*Definition 8. *Let be a meromorphic function of order . A direction is called a Borel direction of order of if, no matter how small the positive number is, for each value , one has
except for at most two exceptional values . A direction is called a Bloch direction of order of if, for any number , any system of distinct values, and any system such that is a positive integer or and that
there exists at least one integer such that

Theorem 9. *Let be a meromorphic function of order . Then every Borel direction of order of is a Bloch direction of order of .*

In order to prove Theorem 9, we need the following lemma.

Lemma 10 (Zhang [7]). *Let be a meromorphic function of order . Then a direction is a Borel direction of order of if and only if it satisfies
**
for any .*

We are now in the position to prove Theorem 9.

*Proof. *Suppose that is a Borel direction of order of ; then, for any , we have
If is not a Bloch direction of order of , then there exit a system of distinct values and a system such that is a positive integer or and that
And, for any integer , we have
Hence, we can get
for any integer . Therefore, we can find a positive number such that
By (39), we have
where Hence,
This contradicts with (48) and Theorem 9 follows.

Corollary 11. *Let be a meromorphic function of order . Then there is a direction which is a Bloch direction of order of .*

Note that Corollary 11 is a corresponding result of Theorem A in angular distribution.

#### 5. Pseudo-T Direction of Meromorphic Functions

In 2003, Zheng [1] introduced a new singular direction, called T direction. We call the T direction of , provided that, given any , possibly with exception of at most two values of , for any positive number , we have For the existence of T direction of meromorphic function , Guo et al. [10] proved the following Theorem.

Theorem C. *Let be a meromorphic function and satisfy
**
Then must have a T direction.*

Theorem C was conjectured by Zheng [1]. In [11], the authors study the existence of T direction of concerning multiple values. We call the T direction of concerning multiple values, provided that, given any , possibly with exception of at most values of , for any positive number , we have where implies the maximum integer number which does not exceed and is a positive integer.

Theorem D. *Let be a meromorphic function and satisfy (56). Then there at least exists a T direction of concerning multiple values.*

Note that the T direction of meromorphic function concerning multiple values is a refinement of the ordinary T direction since as . Since Zheng [1] gave the definition of T direction, then there is a considerable number result related this direction, we refer the reader to [12] for finding a careful discussion of this direction.

It is well known that T direction is a concept in angular distribution which corresponds to the Nevanlinna second fundamental theorem in module distribution. It is natural to consider the corresponding result to Theorem B in angular distribution.

*Definition 12. *Let be a meromorphic function. A direction is called a pseudo-T direction of if, for any number , any system of distinct values, and any system such that is a positive integer or and that
there exists at least one integer such that

Theorem 13. *Let be a meromorphic function and satisfy (56). Then there at least exists a pseudo-T direction of .*

*Remark 14. *(i) In Theorem C, , , so Theorem C is a special case of Theorem 13.

(ii) If , then ; if , then ; if , then . So Theorem D is a special case of Theorem 13.

In order to prove Theorem 13, we need the following lemma.

Lemma 15 (Li and Gu [13], see also Xuan [14]). *Suppose that is a nonnegative increasing function in and satisfies
**
Then for any set such that , one has
*

*Proof. *Firstly, we prove the following statement. Let be a fixed positive integer, . We put , , . Then among these angular domains , there is at least an angular domain such that for any system of distinct values and any system such that is a positive integer or and that
there exists at least one integer such that
Otherwise, for any angular domain , there is a system of distinct values and a system such that is a positive integer or and that
for any we have
Put
Applying Theorem 5 to , we have
Noting and adding two sides of the above expression from to , we can obtain
For any , there exists a , the inequality would bold for , while the inequality (22) does not look appropriate here. Put is the set of which consists of a series of intervals and satisfies
Let ; we have for any , ; then
Applying Lemma 15, we have
where . Therefore, there exists a sequence , such that
It follows from (38), (68), and (72) that
Hence
This is a contradiction. Hence, for an arbitrary positive integer , there is at least an angular domain such that for any system of distinct values and any system such that is a positive integer or and that
there exists at least one integer such that
Choosing subsequence of , still denote it , we assume that . Put ; then is a pseudo-T direction that is stated in Definition 12.

In fact, for any , when is sufficiently large, we have . By (76), we have
Hence, Theorem 13 holds in this case.

#### Conflict of Interests

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

#### Acknowledgments

The first author was partly supported by the National Natural Science Foundation of China (Grant no. 11201395) and by the Science Foundation of Educational Commission of Hubei Province (Grant no. Q20132801). The second author was partly supported by the NSF of Jiangxi Province (Grant 20122BAB201006). The third author was partly supported by Beijing Natural Science Foundation (Grant no. 1132013).