An Inequality of Meromorphic Functions and Its Application

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 f, namely a Bloch direction and a pseudo-T direction for f.


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 C onto a sector containing a single ray : arg = 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.
In [2], Yang has proved the following theorems.
In this paper, we will research the singular directions corresponding to Theorems A and B.

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 0 , consisting of a finite number of sheets, and be bounded by a finite number of analytic Jordan curves {Λ } (some of which may reduce to single points), and the spherical distance between any two circular curves Λ and Λ is (Λ , Λ ) ≥ ∈ (0, 1/2). The part of the boundary of , which does not lie above the boundary of 0 , is called the relative boundary of and denote its spherical length by . Let be a domain on 0 , whose boun-dary 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 0 by | |, | ( )| and | 0 |, 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 0 , one has where ℎ > 0 is a constant which depends on 0 only.
Lemma 2 (see Sun [5]). Let be a simply connected finite covering surface of the unite sphere , and let { V } be (> 2) disjoint spherical disks on , where the spherical distance of any pair of { V } is at least . Let V be the number of simply connected islands (see Tsuji [3,Page 252]) in ( V )); 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 V (V = 1, 2, . . . , ) be positive integers. Let V (V = 1, 2, . . . , ) be (> 2) disjoint spherical disks with radius /3 on and without a pair of { V } such that their spherical distance is less than and let V ) V be the number of simply connected islands in ( V ), which consisted of no more than V sheets; then where is the length of the relative boundary of .
Proof. It is easy to verify that where ( V V is the number of simply connected islands in ( V ), which consist of no less than V + 1 sheets. Hence, Since the spherical area of V is | V | ≥ 2 /9, it follows from Lemma 1 that Note that 1/( V + 1) < 1 and 0 < < 1/2; 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.

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 C; we denote The Scientific World Journal 3 When = C, we write ( , C, ) by 0 ( , ). Then from Theorem 1.4 in [6], we have And the difference ( , ) − 0 ( , ) is a bounded function of , so that both the characteristic function 0 ( , ) and ( , ) are interchangeable. Denote the following angular domain by When is a sector { ∈ C, | | < } ∩ Ω( , ), we denote ( , ) = ( , Ω( , ), ) and For any ∈ C ∞ and ̸ = ∞, let ( , , , ) be the number of zeros, counted according to their multiplicities, of ( ) − in the sector { ∈ C, | | < } ∩ Ω( , ), and let ) ( , , , ) be the number of zeros with multiplicities ≤ , of ( ) − in the sec- 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
V (V = 1, 2, . . . , ) be positive integers, and { V } are (> 2) distinct points on and without a pair of { V } such that their spherical distance is less than V be the number of zeros of ( )− V , which are consisted of not more than V multiplicities, then Proof. Let V be a spherical disk with the center V with radius /3 on . By Theorem 3, we have (20) , whenever V in the island of V or in the peninsula of V . Therefore, Lemma 4 follows.
We are now in the position to establish the main result in this section.   The Scientific World Journal 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 0 > 0, such that ( , ) > 0, when > 0 , and ( , ) ≤ 0, when ≤ 0 . For > 0 , by (25) and (27), that is, Integrating each side of the inequality leads to Thus On the case of ≤ 0 , the above inequality is obviously valid because of ( , ) ≤ 0. Replacing ( , ) in the above The Scientific World Journal 5 inequality with its explicit expression, we see that (21) is established. Therefore where ( , − , + ) = ∫ 1 ( ( , − , + )/ ) .
with at most one exceptional set of , where consists of a series of intervals and satisfies In particular, if the order of ( ) is (0 < < +∞), then From Theorem 3 and Lemma 6, we can write the result in Theorem 3 as If the order of ( ) is (0 < < +∞), then the inequality will be

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 ≥ 0 ( 0 > 0) and tends to +∞ as → +∞.
For a meromorphic function of infinite order, Zhuang Qitai (or Chuang Chitai) [9] gives the following definition of Borel direction and Bloch direction.
except for at most two exceptional values . A direction arg = is called a Bloch direction of order ( ) of ( ) if, for any number (0 < < /2), any system ( = 1, 2, . . . , ) of distinct values and, any system ( = 1, 2, . . . , ) such that is a positive integer or +∞ and that there exists at least one integer (1 ≤ ≤ ) such that lim sup → ∞ log ) ( , , , ) 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 ( ).

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The Scientific World Journal It is natural to consider whether there exists a similar result, if meromorphic function of order infinity is replaced with meromorphic function of order (0 < < +∞). In this section we extend the above theorem to meromorphic function of order (0 < < +∞).
Definition 8. Let ( ) be a meromorphic function of order (0 < < +∞). A direction arg = is called a Borel direction of order of ( ) if, no matter how small the positive number is, for each value , one has lim sup except for at most two exceptional values . A direction arg = is called a Bloch direction of order of ( ) if, for any number (0 < < /2), any system ( = 1, 2, . . . , ) of distinct values, and any system ( = 1, 2, . . . , ) such that is a positive integer or +∞ and that there exists at least one integer (1 ≤ ≤ ) such that (46) Theorem 9. Let ( ) be a meromorphic function of order (0 < < +∞). 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.
We are now in the position to prove Theorem 9.
Proof. Suppose that arg = is a Borel direction of order of ( ); then, for any (0 < < /2), we have lim sup If arg = is not a Bloch direction of order of ( ), then there exit a system ( = 1, 2, . . . , ) of distinct values and a system ( = 1, 2, . . . , ) such that is a positive integer or +∞ and that And, for any integer (1 ≤ ≤ ), we have lim sup Hence, we can get lim sup for any integer (1 ≤ ≤ ). Therefore, we can find a positive number < such that By (39), we have This contradicts with (48) and Theorem 9 follows.
Corollary 11. Let ( ) be a meromorphic function of order (0 < < +∞). Then there is a direction arg = which is a Bloch direction of order of ( ).
Note that Corollary 11 is a corresponding result of Theorem A in angular distribution.

Pseudo-T Direction of Meromorphic Functions
In 2003, Zheng [1] introduced a new singular direction, called T direction. We call : arg = the T direction of ( ), provided that, given any ∈ C ∞ , possibly with exception of at most two values of , for any positive number < , we have Then ( ) must have a T direction.
The Scientific World Journal 7 Theorem C was conjectured by Zheng [1]. In [11], the authors study the existence of T direction of ( ) concerning multiple values. We call : arg = the T direction of ( ) concerning multiple values, provided that, given any ∈ C ∞ , possibly with exception of at most [(2 + 2)/ ] 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 [(2 + 2)/ ] → 2 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.