Research ArticleAn Inequality of Meromorphic Functions and Its Application

Zhaojun Wu1 Yuxian Chen2 and Zuxing Xuan3

1 School of Mathematics and Statistics Hubei University of Science and Technology Xianning 437100 China2 School of Mathematics and Computer Science Xinyu University Xinyu 338004 China3 Beijing Key Laboratory of Information Service Engineering Department of General Education Beijing Union UniversityNo 97 Bei Si Huan Dong Road Chaoyang District Beijing 100101 China

Correspondence should be addressed to Zhaojun Wu wuzj52hotmailcom

Received 31 December 2013 Accepted 22 January 2014 Published 6 March 2014

Academic Editors G M Amiraliyev and F Kittaneh

Copyright copy 2014 Zhaojun Wu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

By applying Ahlfors theory of covering surface we establish a fundamental inequality of meromorphic function dealing withmultiple values in an angular domain As an application we prove the existence of some new singular directions for a meromorphicfunction 119891 namely a Bloch direction and a pseudo-T direction for 119891

1 Introduction

In this papermeromorphic function alwaysmeans a functionmeromorphic in thewhole complex plane Given ameromor-phic function 119891(119911) the theory of value distribution of 119891(119911)developed in the twoways one is themodule distribution andthe other is angular distribution For the module distributionof a meromorphic function there are three main theoremsthat is the Picard theorem the Borel theorem and theNevanlinna second fundamental theorem The fundamentalconcept in the angular distribution is singular direction Sin-gular direction is a concept of localizing value distribution inC onto a sector 119878 containing a single ray 119869 arg 119911 = 120579emanating from the origin say A Julia direction and a Boreldirection are refinements of the Picard theorem and the Boreltheorem respectively Corresponding to the Nevanlinna sec-ond fundamental theorem a new singular direction calledT direction was recently introduced in Zheng [1]Whenmul-tiple values were considered Yang [2] proved the followingtheorems related to the module distribution of meromorphicfunction In order to introduce the main results of Yang wegive some notations (see [2]) as the following

Let 119891(119911) denote a nonconstant meromorphic function119886 isin C an arbitrary complex number and 119896 a positive integerWe use 119899119896)(119903 1(119891 minus 119886)) or 119899119896)(119903 119886) to denote the zeros of

119891(119911) minus 119886 in |119911| le 119903 whose multiplicities are no greater than119896 counted according to their multiplicities Likewise we use119899(119896(119903 1(119891 minus 119886)) or 119899(119896(119903 119886) to denote those zeros in |119911| le 119903whose multiplicities are greater than 119896 counted according totheir multiplicities The corresponding counting functionsare denoted by119873119896)(119903 1(119891minus119886)) or119873119896)(119903 119886) and119873(119896(119903 1(119891minus119886)) or 119873(119896(119903 119886) Let 119891(119911) be a meromorphic function withorder 120588 (0 lt 120588 lt +infin) 119886 be an arbitrary number and 119896 be apositive integer If

lim sup119903rarrinfin

log 119899119896) (119903 119886)log 119903

lt 120588 (1)

then 119886 is called a pseudo-Borel exceptional value of 119891(119911) oforder 119896

In [2] Yang has proved the following theorems

Theorem A Let 119891(119911) be a meromorphic function with order120588 (0 lt 120588 lt +infin) and let 119896119895 (119895 = 1 2 119902) be 119902 positive inte-gers If119891(119911) has 119902 distinct pseudo-Borel exceptional values 119886119895 oforder 119896119895 (119895 = 1 2 119902) then

119902

sum119895=1

(1 minus1

119896119895 + 1) le 2 (2)

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 242851 9 pageshttpdxdoiorg1011552014242851

2 The Scientific World Journal

Theorem B Let 119891(119911) be a nonconstant meromorphic 119886119895 isinCinfin (119895 = 1 2 119902) be 119902(ge 3) distinct complex numbers and119896119895 (119895 = 1 2 119902) be 119902 positive integers Then

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 119891)

lt

119902

sum119895=1

119896119895

119896119895 + 1119873119896119895) (119903

1

119891 minus 119886119895) + 119878 (119903 119891)

(3)

where 119878(119903 119891) is the Nevanlinna error term

In this paper we will research the singular directions cor-responding toTheorems A and B

2 A Theorem on Covering Surface

In this section wewill give a theoremon covering surfaceWefirstly introduce the following notations (see Tsuji [3])

In this paper theRiemann sphere of diameter 1 is denotedby 119870 Let 119865 be a finite covering surface of 1198650 consisting of afinite number of sheets and be bounded by a finite number ofanalytic Jordan curves Λ119895 (some of which may reduce tosingle points) and the spherical distance between any two cir-cular curves Λ119894 and Λ119895 is 119889(Λ 119894 Λ 119895) ge 120575 isin (0 12) The partof the boundary of 119865 which does not lie above the boundaryof1198650 is called the relative boundary of119865 and denote its spher-ical length by 119871 Let 119863 be a domain on 1198650 whose boun-daryconsists a finite number of points or analytic closed Jordancurves and let 119865(119863) be the part of 119865 which lies above 119863We denote the spherical area of 119865 119865(119863) and 1198650 by |119865| |119865(119863)|and |1198650| respectively We put

119878 =|119865|10038161003816100381610038161198650

1003816100381610038161003816 119878 (119863) =

|119865 (119863)|

|119863| (4)

Under the above notation we have the following Ahlfors cov-eringTheorem

Lemma1 (seeTsuji [3]) For any finite covering surface119865 of1198650one has

|119878 minus 119878 (119863)| lt ℎ119871

|119863| (5)

where ℎ gt 0 is a constant which depends on 1198650 only

Recently Sun [4] has proved a precise version of Lemma 1and proved that ℎ = 2120587120575 where 0 lt 120575 lt 12 is a constant

Lemma 2 (see Sun [5]) Let 119865 be a simply connected finitecovering surface of the unite sphere 119870 and let 119863V be 119902(gt 2)disjoint spherical disks on119870 where the spherical distance of anypair of 119863V is at least 120575 Let 119899V be the number of simply con-nected islands (see Tsuji [3 Page 252]) in 119865(119863V)) then

119902

sumV=1119899V ge (119902 minus 2) 119878 minus

119862

1205753119871 (6)

where 119871 is the length of the relative boundary of 119865 and 119862 is aconstant

Theorem3 Let119865 be a simply connected finite covering surfaceof the unite sphere 119870 and let 119897V (V = 1 2 119902) be 119902 positiveintegers Let 119863V (V = 1 2 119902) be 119902(gt 2) disjoint sphericaldisks with radius 1205753 on119870 and without a pair of 119863V such thattheir spherical distance is less than 120575 and let 119899119897V)V be the numberof simply connected islands in 119865(119863V) which consisted of nomore than 119897V sheets then

119902

sumV=1

119897V119897V + 1

119899119897V)V ge (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878

minus119862 + 9119902ℎ

1205753119871

(7)

where 119871 is the length of the relative boundary of 119865

Proof It is easy to verify that

119899V = 119899119897V)V + 119899(119897VV 119878 (119863V) ge 119899119897V)V + (119897 + 1) 119899(119897VV (8)

where 119899(119897VV is the number of simply connected islands in119865(119863V)which consist of no less than 119897V + 1 sheets Hence

119878 (119863V) ge (119897V + 1) (119899119897V)V + 119899(119897VV ) minus 119897V119899

119897V)V = (119897V + 1) 119899V minus 119897V119899

119897V)V

(9)

Since the spherical area of 119863V is |119863V| ge 12057529 it follows fromLemma 1 that

119878 +9ℎ

1205752119871 gt 119878 (119863V) ge (119897V + 1) 119899V minus 119897V119899

119897V)V (10)

Note that 1(119897V + 1) lt 1 and 0 lt 120575 lt 12 we can get

119899V le119897V

119897V + 1119899119897V)V +

1

119897V + 1119878 +

9ℎ

1205753119871 (11)

Adding two sides of the above expression from 1 to 119902 we have

119902

sumV=1119899V le

119902

sumV=1

119897V119897V + 1

119899119897V)V +

119902

sumV=1

1

119897V + 1119878 +

9119902ℎ

1205753119871 (12)

Combining Lemma 2 and the above expression Theorem 3follows

3 A Fundamental Inequality of MeromorphicFunctions in an Angular Domain

TheAhlfors-Shimizu characteristic is important in this paperLet us recall its definition Suppose that 119864 is a nonempty sub-set of C we denote

119878 (119903 119864 119891) =1

120587intint119864

(

100381610038161003816100381610038161198911015840(119911)

10038161003816100381610038161003816

1 +1003816100381610038161003816119891(119911)

10038161003816100381610038162)

2

119889119908

1198790 (119903 119864 119891) = int119903

0

119878 (119905 119864 119891)

119905119889119905

(13)

The Scientific World Journal 3

When 119864 = C we write 119879(119903C 119891) by 1198790(119903 119891) Then fromTheorem 14 in [6] we have

1003816100381610038161003816119879 (119903 119891) minus log+ 1003816100381610038161003816119891 (0)1003816100381610038161003816 minus 1198790 (119903 119891)

1003816100381610038161003816 le1

2log 2 (14)

And the difference 119879(119903 119891)minus1198790(119903 119891) is a bounded function of119903 so that both the characteristic function1198790(119903 119891) and119879(119903 119891)are interchangeable Denote the following angular domain by

Ω (120579 120576) = 119911 isin C1003816100381610038161003816arg 119911 minus 120579

1003816100381610038161003816 lt 120576 (15)

When 119864 is a sector 119911 isin C |119911| lt 119903 cap Ω(120579 120576) we denote119878(119864 119891) = 119878(119903 Ω(120579 120576) 119891) and

119879 (119903 Ω (120579 120576) 119891) = int119903

0

119878 (119905 Ω (120579 120576) 119891)

119905119889119905 (16)

For any 119886 isin Cinfin and 119886 =infin let 119899(119903 120579 120576 119886) be the number ofzeros counted according to their multiplicities of 119891(119911)minus119886 inthe sector 119911 isin C |119911| lt 119903capΩ(120579 120576) and let 119899119897)(119903 120579 120576 119886) be thenumber of zeros with multiplicities le 119897 of 119891(119911)minus119886 in the sec-tor 119911 isin C |119911| lt 119903 cap Ω(120579 120576) where 119897 is any positive integerSimilarly note the number of poles of 119891 by 119899(119903 120579 120576infin) and119899119897)(119903 120579 120576infin) Denote

119873(119903 120579 120576 119886) = int119903

0

119899 (119905 120579 120576 119886) minus 119899 (0 120579 120576 119886)

119905119889119905

+ 119899 (0 120579 120576 119886) log 119903

119873119897) (119903 120579 120576 119886) = int119903

0

119899119897) (119905 120579 120576 119886) minus 119899119897) (0 120579 120576 119886)

119905119889119905

+ 119899119897) (0 120579 120576 119886) log 119903

(17)

In addition we also need the notations (see [7])

119871 (119903 1205951 1205952) = int1205952

1205951

100381610038161003816100381610038161198911015840(119903119890119894120595)

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119903119890119894120595)10038161003816100381610038162)119903 119889120595

119871 (119903 120595) = int119903

1

100381610038161003816100381610038161198911015840(119905119890119894120595)

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894120595)10038161003816100381610038162)119889119905

(18)

In this section we will establish a fundamental inequalityfor meromorphic functions in an angular domain Firstly wegive the following lemma

Lemma 4 Suppose that 119891(119911) is a meromorphic function and119897V (V = 1 2 119902) be 119902 positive integers and 119886V are 119902(gt 2)distinct points on 119870 and without a pair of 119886V such that theirspherical distance is less than 120575 + 21205753 119899119897V)V be the number ofzeros of119891(119911)minus119886V which are consisted of notmore than 119897V multi-plicities then

119902

sumV=1

(119897V

119897V + 1) 119899119897V)V ge (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 minus

119862 + 9119902ℎ

1205753119871

(19)

Proof Let119863V be a spherical diskwith the center 119886V with radius1205753 on119870 By Theorem 3 we have

119902

sumV=1

(119897V

119897V + 1) 119899119897V)V ge (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 minus

119862 + 9119902ℎ

1205753119871

(20)

Note that 119899119897V)V (119863V) le 119899119897V)V (119886V) whenever 119886V in the island of 119863Vor in the peninsula of119863V Therefore Lemma 4 follows

We are now in the position to establish the main result inthis section

Theorem 5 Let 119891(119911) be a meromorphic function and 119897V (V =1 2 119902) 119902 positive integers If 119886V are 119902 distinct points on119870then one has

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V)

+21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log 119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891)

+ 119867119871 (1 120579 minus 120575 120579 + 120575) + 119867119871 (119903 120579 minus 120575 120579 + 120575)

(21)

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log2119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (1Ω (120579 120593) 119891)

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891) log 119903

+ 119867119871 (1 120579 minus 120575 120579 + 120575) log 119903 + 120594 (119903 120579 minus 120575 120579 + 120575)(22)

for any 120593 0 lt 120593 lt 120575 where119867 is a constant depending only on119886V V = 1 2 119902 and 120594(119903 120579 minus 120575 120579 + 120575) = 119867int

119903

1(119871(119905 120579 minus 120575 120579 +

120575)119905)119889119905

4 The Scientific World Journal

Proof Put 119863119903 = 119911 isin C 1 lt |119911| lt 119903 cap Ω(120579 120593) and 1198650 = 119870 minus119886V Using Lemma 4 we have

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2) [119878 (119903 Ω (120579 120593) 119891)

minus119878 (1 Ω (120579 120593) 119891)]

le

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V) + 119867119871 (119903)

(23)

where119867 = (119862 + 9119902ℎ)1198971205753 which depends only on 1198650 that isonly on 119886V V = 1 2 119902 and

119871 (119903) = 119871 (119903 120579 minus 120593 120579 + 120593) + 119871 (1 120579 minus 120593 120579 + 120593)

+ 119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)

le 119871 (119903 120579 minus 120575 120579 + 120575) + 119871 (1 120579 minus 120575 120579 + 120575)

+ 119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)

(24)

Hence

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)

times [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

minus

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V) minus 119867119871 (119903 120579 minus 120575 120579 + 120575)

minus 119867119871 (1 120579 minus 120575 120579 + 120575)

le 119867 [119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]

(25)

Denote the left expression of (25) by 119860(119903 120593) thus

119889 (119860 (119903 120593))

119889120593

= (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)

times119889 [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

119889120593

(26)

We claim the fact that

[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]2le

2120587

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

times119889 (119860 (119903 120593))

119889120593log 119903

(27)

In fact it follows from the definition of 119871(119903 120595) and Schwarzrsquosinequality that

[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]2

le 2[

[

(int119903

1

100381610038161003816100381610038161198911015840(119905119890119894(120579minus120593))

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894(120579minus120593))

10038161003816100381610038162)119889119905)

2

+ (int119903

1

100381610038161003816100381610038161198911015840(119905119890119894(120579+120593))

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894(120579+120593))

10038161003816100381610038162)119889119905)

2

]

]

le 2120587119889 [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

119889120593log 119903

=2120587

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

119889 (119860 (119903 120593))

119889120593log 119903

(28)

Noting119860(119903 120593) is an increasing function of 120593 we see that thenthere exists a 1205750 gt 0 such that 119860(119903 120593) gt 0 when 120593 gt 1205750 and119860(119903 120593) le 0 when 120593 le 1205750 For 120593 gt 1205750 by (25) and (27)

[119860 (119903 120593)]2le 1198672[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]

2

le21205871198672

(sum119902

119895=1(1 minus

1

119897V + 1) minus 2)

log 119903119889 (119860 (119903 120593))

119889120593

(29)

that is

119889120593 le21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

log 119903119889 (119860 (119903 120593))

[119860 (119903 120593)]2 (30)

Integrating each side of the inequality leads to

120575 minus 120593 = int120575

120593

119889120593 le21205871198672

(sum119902

119895=1(1 minus

1

119897V + 1) minus 2)119860 (119903 120593)

log 119903

(31)

Thus

119860 (119903 120593) le21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120591)

log 119903 (32)

On the case of 120593 le 1205750 the above inequality is obviously validbecause of 119860(119903 120593) le 0 Replacing 119860(119903 120593) in the above

The Scientific World Journal 5

inequality with its explicit expression we see that (21) is esta-blished Therefore

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+1205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log2119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (1Ω (120579 120593) 119891)

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891) log 119903

+ 119867119871 (1 120579 minus 120575 120579 + 120575) log 119903 + 120594 (119903 120579 minus 120575 120579 + 120575) (33)

where 120594(119903 120579 minus 120575 120579 + 120575) = 119867int119903

1(119871(119905 120579 minus 120575 120579 + 120575)119905)119889119905

Lemma 6 (Zhang [7]) Under the condition ofTheorem 5 onehas

120594 (119903 120579 minus 120575 120579 + 120575) = 119867int119903

1

119871 (119905 120579 minus 120575 120579 + 120575)

119905119889119905

le 119867radic2120575120587119878 (119903 Ω (120579 120575) 119891) log 119903(34)

or

120594 (119903 120579 minus 120575 120579 + 120575) le 119867radic2120575120587119879 (119903 Ω (120579 120575) 119891)

times log119879 (119903 Ω (120579 120575) 119891)

(35)

with at most one exceptional set 119864120575 of 119903 where 119864120575 consists of aseries of intervals and satisfies

int119864120575

1

119903 log 119903119889119903 le

1

log119879 (119903 Ω (120579 120575) 119891)lt infin (36)

In particular if the order of 119891(119911) is 120588 (0 lt 120588 lt +infin) then

120594 (119903 120579 minus 120575 120579 + 120575) le 119874 (11990331205884) (37)

FromTheorem 3 and Lemma 6 we can write the result inTheorem 3 as

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+ 119874 (log2119903) + 120594 (119903 120579 minus 120575 120579 + 120575)

(38)

If the order of119891(119911) is120588 (0 lt 120588 lt +infin) then the inequalitywillbe

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V) + 119874 (11990331205884)

(39)

4 Bloch Direction of Meromorphic Functions

In this section we will research the singular direction corre-sponding toTheoremA Suppose that119891(119911) is a meromorphicfunction of infinite order Then there is a real function 120588(119903)called anHiongrsquos proximate order (see [8]) of119891(119911) which hasthe following properties (i) 120588(119903) is continuous and nonde-creasing for 119903 ge 1199030 (1199030 gt 0) and tends to +infin as 119903 rarr +infin (ii)The function 119880(119903) = 119903120588(119903)(119903 ge 1199030) satisfies the condition

lim119903rarr+infin

log119880 (119877)

log119880 (119903)= 1 119877 = 119903 +

119903

log119880 (119903)

lim sup119903rarr+infin

log119879 (119903 119891)log119880 (119903)

= 1

(40)

For a meromorphic function of infinite order ZhuangQitai (or Chuang Chitai) [9] gives the following definition ofBorel direction and Bloch direction

Definition 7 Let 119891(119911) be a meromorphic function of infiniteorder and 120588(119903) an order of119891(119911) A direction arg 119911 = 120579 is calleda Borel direction of order 120588(119903) of 119891(119911) if no matter how smallthe positive number 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)120588 (119903) log 119903

= 1 (41)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588(119903) of119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that119896119895 is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (42)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)120588 (119903) log 119903

= 1 (43)

For the connection of Borel direction and Bloch directionof meromorphic function of infinite order Chuang [9] hasproved the following theorem

Theorem C Let 119891(119911) be a meromorphic function of infiniteorder and 120588(119903) an order of 119891(119911) Then every Borel direction oforder 120588(119903) of 119891(119911) is a Bloch direction of order 120588(119903) of 119891(119911)

6 The Scientific World Journal

It is natural to consider whether there exists a similarresult if meromorphic function of order infinity is replacedwithmeromorphic function of order 120588 (0 lt 120588 lt +infin) In thissection we extend the above theorem to meromorphic func-tion of order 120588 (0 lt 120588 lt +infin)

Definition 8 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) A direction arg 119911 = 120579 is called a Borel dir-ection of order 120588 of 119891(119911) if no matter how small the positivenumber 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)log 119903

= 120588 (44)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588 of 119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (45)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)log 119903

= 120588 (46)

Theorem 9 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then every Borel direction of order 120588 of 119891(119911)is a Bloch direction of order 120588 of 119891(119911)

In order to prove Theorem 9 we need the followinglemma

Lemma 10 (Zhang [7]) Let 119891(119911) be a meromorphic functionof order 120588 (0 lt 120588 lt +infin) Then a direction arg 119911 = 120579 is a Boreldirection of order 120588 of 119891(119911) if and only if it satisfies

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (47)

for any 120576 (0 lt 120576 lt 1205872)

We are now in the position to proveTheorem 9

Proof Suppose that arg 119911 = 120579 is a Borel direction of order 120588 of119891(119911) then for any 120576 (0 lt 120576 lt 1205872) we have

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (48)

If arg 119911 = 120579 is not a Bloch direction of order 120588 of 119891(119911) thenthere exit a system 119886119895 (119895 = 1 2 119902) of distinct values and asystem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (49)

And for any integer 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (50)

Hence we can get

lim sup119903rarrinfin

log119873119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (51)

for any integer 119895 (1 le 119895 le 119902)Therefore we can find a positivenumber 120591 lt 120588 such that

119873119896119895) (119903 120579 2120576 119886119895) le 119903120591 (52)

By (39) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Ω (120579 120576) 119891)

le

119902

sumV=1

119896119895

119896119895 + 1119873119896119895) (119903 120579 2120576 119886V) + 119874 (11990331205884)

le 119874 (119903120577)

(53)

where 120577 = max120591 31205884 lt 120588Hence

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120577 lt 120588 (54)

This contradicts with (48) andTheorem 9 follows

Corollary 11 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then there is a direction arg 119911 = 120579 which is aBloch direction of order 120588 of 119891(119911)

Note that Corollary 11 is a corresponding result of Theo-rem A in angular distribution

5 Pseudo-T Direction ofMeromorphic Functions

In 2003 Zheng [1] introduced a new singular direction calledT direction We call 119869 arg 119911 = 120579 the T direction of 119891(119911)provided that given any 119886 isin Cinfin possiblywith exception of atmost two values of 119886 for any positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873(119903 120579 120576 119886)

119879 (119903 119891)gt 0 (55)

For the existence of T direction of meromorphic function119891(119911) Guo et al [10] proved the followingTheorem

Theorem C Let 119891(119911) be a meromorphic function and satisfy

lim sup119903rarrinfin

119879 (119903 119891)

log2119903= infin (56)

Then 119891(119911)must have a T direction

The Scientific World Journal 7

Theorem C was conjectured by Zheng [1] In [11] theauthors study the existence of T direction of 119891(119911) concerningmultiple values We call 119869 arg 119911 = 120579 the T direction of 119891(119911)concerningmultiple values provided that given any 119886 isin Cinfinpossibly with exception of at most [(2119897 + 2)119897] values of 119886 forany positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873119897) (119903 120579 120576 119886)

119879 (119903 119891)gt 0 (57)

where [119909] implies the maximum integer number which doesnot exceed 119909 and 119897 is a positive integer

Theorem D Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a T direction of 119891(119911) concerningmultiple values

Note that the T direction of meromorphic function con-cerning multiple values is a refinement of the ordinary T dir-ection since [(2119897+2)119897] rarr 2 as 119897 rarr infin Since Zheng [1] gavethe definition of T direction then there is a considerablenumber result related this directionwe refer the reader to [12]for finding a careful discussion of this direction

It is well known that T direction is a concept in angulardistribution which corresponds to the Nevanlinna secondfundamental theorem in module distribution It is natural toconsider the corresponding result to Theorem B in angulardistribution

Definition 12 Let 119891(119911) be a meromorphic function A direc-tion arg 119911 = 120579 is called a pseudo-T direction of119891(119911) if for anynumber 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (58)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 120579 120576 119886119895)

119879 (119903 119891)gt 0 (59)

Theorem 13 Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a pseudo-T direction of 119891(119911)

Remark 14 (i) In Theorem C 119902 = 3 119896119895 = infin (119895 = 1 2 3) soTheorem C is a special case of Theorem 13

(ii) If 119896119895 = 1 (119895 = 1 2 119902) then 119902 = 5 if 119896119895 = 2 (119895 =1 2 119902) then 119902 = 4 if 119896119895 = 119897 ge 3 (119895 = 1 2 119902) then119902 = 3 So Theorem D is a special case of Theorem 13

In order to prove Theorem 13 we need the followinglemma

Lemma 15 (Li and Gu [13] see also Xuan [14]) Suppose thatΨ(119903) is a nonnegative increasing function in (1infin) and satisfies

lim sup119903rarrinfin

Ψ (119903)

log2119903= infin (60)

Then for any set 119864 sub (1infin) such that int119864(1119903 log 119903)119889119903 lt 13

one has

lim sup119903rarrinfin119903isin(1infin)minus119864

Ψ (119903)

log2119903= infin (61)

Proof Firstly we prove the following statement Let 119898 (119898 ge4) be a fixed positive integer 1205790 = 0 1205791 = 2120587119898 120579119898minus1 =(119898 minus 1)2120587119898 120579119898 = 1205790 We put Δ(120579119894) = 119911 | arg 119911 minus 120579119894| lt2120587119898 Δ119900(120579119894) = 119911 | arg 119911 minus 120579119894| lt 120587119898 119894 = 0 1 119898 minus 1Δ(120579119898) = Δ(1205790) Δ

119900(120579119898) = Δ119900(1205790) Then among these119898 angu-lar domains Δ(120579119894) there is at least an angular domain Δ(120579119894)such that for any system 119886119895 (119895 = 1 2 119902) of distinct valuesand any system 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positiveinteger or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (62)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (63)

Otherwise for any angular domainΔ(120579119894) (1 le 119894 le 119898) there isa system 119886

119895

119894(119895 = 1 2 119902) of distinct values and a system

119896119895

119894(119895 = 1 2 119902) such that 119896119895

119894is a positive integer or+infin and

that

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

) gt 2 (64)

for any 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

119873119896119895

119894) (119903 Δ (120579119894) 119886

119895

119894)

119879 (119903 119891)= 0 (65)

Put

119902

sum119895=1

(1 minus1

119896119895 + 1) = min1le119894le119898

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

)

gt 2 (66)

ApplyingTheorem 5 to Δ119900(120579119894+1) Δ(120579119894+1) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Δ119900 (120579119894+1) 119891)

le

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895

119894) (119903 Δ (120579119894+1) 119886

119895

119894+1)

+ 119874 (log2119903) + 120594 (119903 Δ (120579119894+1))

(67)

8 The Scientific World Journal

Noting119879(119903 119891) = sum119898minus1

119894=0119879(119903 Δ119900(120579119894+1) 119891) and adding two sides

of the above expression from 119894 = 0 to119898 minus 1 we can obtain

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 119891)

le119898minus1

sum119894=0

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895) (119903 Δ (120579119894+1) 119886119895

119894+1)

+ 119874 (log2119903) +119898minus1

sum119894=0

120594 (119903 Δ (120579119894+1))

(68)

For any 119894 there exists a 119903119894 the inequality 119879(119903 Δ119900(120579119894+1) 119891) gt

1198903119898 would bold for 119903 gt 119903119894 while the inequality (22) does notlook appropriate here Put 119864Δ119900(120579119894+1) is the set of 119903 which con-sists of a series of intervals and satisfies

int119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt

1

3119898 (69)

Let 1199030 = max119903119894 119894 = 1 2 119898 we have for any 119894 119879(1199030Δ119900(120579119894+1) 119891) gt 1198903119898 then

intcup119898minus1119894=0119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le119898minus1

sum119894=0

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt1

3

(70)

Applying Lemma 15 we have

lim sup119903rarrinfin119903isin(1infin)minus119864

119879 (119903 119891)

log2119903= infin (71)

where 119864 = cup119898minus1119894=0

119864Δ119900(120579119894+1) Therefore there exists a sequence1199031015840119899isin (1infin) minus 119864 such that

lim119899rarrinfin

119879 (1199031015840119899 119891)

log21199031015840119899

= infin (72)

It follows from (38) (68) and (72) that

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2) le 0 (73)

Hence119902

sum119895=1

(1 minus1

119896119895 + 1) le 2 (74)

This is a contradiction Hence for an arbitrary positive inte-ger 119898 there is at least an angular domain Δ(120579119894) such that forany system 119886119895 (119895 = 1 2 119902) of distinct values and any sys-tem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (75)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (76)

Choosing subsequence of 120579119898 still denote it 120579119898 we assumethat 120579119898 rarr 1205790 Put 119871 arg 119911 = 1205790 then 119871 is a pseudo-Tdirection that is stated in Definition 12

In fact for any 120576 (0 lt 120576 lt 1205872) when 119898 is sufficientlylarge we have Δ(120579119898) sub Ω(1205790 120576) By (76) we have

lim sup119903rarrinfin

119873119896119895) (119903 1205790 120576 119886119895)

119879 (119903 119891)ge lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119898) 119886119895)

119879 (119903 119891)gt 0

(77)

Hence Theorem 13 holds in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The first author was partly supported by the National NaturalScience Foundation of China (Grant no 11201395) and by theScience Foundation of Educational Commission of HubeiProvince (Grant no Q20132801) The second author waspartly supported by the NSF of Jiangxi Province (Grant20122BAB201006) The third author was partly supported byBeijing Natural Science Foundation (Grant no 1132013)

References

[1] J H Zheng ldquoOn transcendental meromorphic functions withradially distributed valuesrdquo Science in China A vol 47 no 3 pp401ndash416 2004

[2] L Yang Value Distribution Theory Springer Berlin Germany1993

[3] M Tsuji PotentialTheory inModern FunctionTheoryMaruzenTokyo Japan 1959

[4] D C Sun ldquoMain theorem on covering surfacesrdquo Acta Mathe-matica Scientia B vol 14 no 2 pp 213ndash225 1994

[5] D C Sun ldquoInequalities for covering surfaces and their applica-tionsrdquoChinese Annals ofMathematics A vol 18 no 1 pp 59ndash641997 (Chinese)

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs Clarendon Press Oxford UK 1964

[7] X L Zhang ldquoA fundamental inequality for meromorphic func-tions in an angular domain and its applicationrdquoActaMathemat-ica Sinica vol 10 no 3 pp 308ndash314 1994

[8] Q T Zhuang Singular Directions of Meromorphic Functionsvol 11 of Series of Monographs in Pure and AppliedMathematicsScience Press Beijing China 1982 (Chinese)

[9] C T Chuang ldquoOn Borel directions of meromorphic functionsof infinite order IIrdquoBulletin of theHongKongMathematical Soc-iety vol 2 no 2 pp 305ndash323 1999

[10] H Guo J H Zheng and TW Ng ldquoOn a new singular directionof meromorphic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 69 no 2 pp 277ndash287 2004

The Scientific World Journal 9

[11] Z-J Wu and D-C Sun ldquoOn the existence of 119879 direction ofmeromorphic function concerning multiple valuesrdquo KodaiMathematical Journal vol 31 no 1 pp 133ndash149 2008

[12] J H Zheng Value Distribution of Meromorphic FunctionsTsinghua University Press Beijing China 2010

[13] C H Li and Y X Gu ldquoA fundamental inequality for 119870-quasimeromorphic mappings in an angular domain and itsapplicationrdquo Acta Mathematica Sinica vol 49 no 6 pp 1279ndash1286 2006 (Chinese)

[14] Z-X Xuan ldquoOn the existence of 119879-direction of algebroidfunctions a problem of J H Zhengrdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 540ndash547 2008

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Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

Theorem B Let 119891(119911) be a nonconstant meromorphic 119886119895 isinCinfin (119895 = 1 2 119902) be 119902(ge 3) distinct complex numbers and119896119895 (119895 = 1 2 119902) be 119902 positive integers Then

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 119891)

lt

119902

sum119895=1

119896119895

119896119895 + 1119873119896119895) (119903

1

119891 minus 119886119895) + 119878 (119903 119891)

(3)

where 119878(119903 119891) is the Nevanlinna error term

In this paper we will research the singular directions cor-responding toTheorems A and B

2 A Theorem on Covering Surface

In this section wewill give a theoremon covering surfaceWefirstly introduce the following notations (see Tsuji [3])

In this paper theRiemann sphere of diameter 1 is denotedby 119870 Let 119865 be a finite covering surface of 1198650 consisting of afinite number of sheets and be bounded by a finite number ofanalytic Jordan curves Λ119895 (some of which may reduce tosingle points) and the spherical distance between any two cir-cular curves Λ119894 and Λ119895 is 119889(Λ 119894 Λ 119895) ge 120575 isin (0 12) The partof the boundary of 119865 which does not lie above the boundaryof1198650 is called the relative boundary of119865 and denote its spher-ical length by 119871 Let 119863 be a domain on 1198650 whose boun-daryconsists a finite number of points or analytic closed Jordancurves and let 119865(119863) be the part of 119865 which lies above 119863We denote the spherical area of 119865 119865(119863) and 1198650 by |119865| |119865(119863)|and |1198650| respectively We put

119878 =|119865|10038161003816100381610038161198650

1003816100381610038161003816 119878 (119863) =

|119865 (119863)|

|119863| (4)

Under the above notation we have the following Ahlfors cov-eringTheorem

Lemma1 (seeTsuji [3]) For any finite covering surface119865 of1198650one has

|119878 minus 119878 (119863)| lt ℎ119871

|119863| (5)

where ℎ gt 0 is a constant which depends on 1198650 only

Recently Sun [4] has proved a precise version of Lemma 1and proved that ℎ = 2120587120575 where 0 lt 120575 lt 12 is a constant

Lemma 2 (see Sun [5]) Let 119865 be a simply connected finitecovering surface of the unite sphere 119870 and let 119863V be 119902(gt 2)disjoint spherical disks on119870 where the spherical distance of anypair of 119863V is at least 120575 Let 119899V be the number of simply con-nected islands (see Tsuji [3 Page 252]) in 119865(119863V)) then

119902

sumV=1119899V ge (119902 minus 2) 119878 minus

119862

1205753119871 (6)

where 119871 is the length of the relative boundary of 119865 and 119862 is aconstant

Theorem3 Let119865 be a simply connected finite covering surfaceof the unite sphere 119870 and let 119897V (V = 1 2 119902) be 119902 positiveintegers Let 119863V (V = 1 2 119902) be 119902(gt 2) disjoint sphericaldisks with radius 1205753 on119870 and without a pair of 119863V such thattheir spherical distance is less than 120575 and let 119899119897V)V be the numberof simply connected islands in 119865(119863V) which consisted of nomore than 119897V sheets then

119902

sumV=1

119897V119897V + 1

119899119897V)V ge (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878

minus119862 + 9119902ℎ

1205753119871

(7)

where 119871 is the length of the relative boundary of 119865

Proof It is easy to verify that

119899V = 119899119897V)V + 119899(119897VV 119878 (119863V) ge 119899119897V)V + (119897 + 1) 119899(119897VV (8)

where 119899(119897VV is the number of simply connected islands in119865(119863V)which consist of no less than 119897V + 1 sheets Hence

119878 (119863V) ge (119897V + 1) (119899119897V)V + 119899(119897VV ) minus 119897V119899

119897V)V = (119897V + 1) 119899V minus 119897V119899

119897V)V

(9)

Since the spherical area of 119863V is |119863V| ge 12057529 it follows fromLemma 1 that

119878 +9ℎ

1205752119871 gt 119878 (119863V) ge (119897V + 1) 119899V minus 119897V119899

119897V)V (10)

Note that 1(119897V + 1) lt 1 and 0 lt 120575 lt 12 we can get

119899V le119897V

119897V + 1119899119897V)V +

1

119897V + 1119878 +

9ℎ

1205753119871 (11)

Adding two sides of the above expression from 1 to 119902 we have

119902

sumV=1119899V le

119902

sumV=1

119897V119897V + 1

119899119897V)V +

119902

sumV=1

1

119897V + 1119878 +

9119902ℎ

1205753119871 (12)

Combining Lemma 2 and the above expression Theorem 3follows

3 A Fundamental Inequality of MeromorphicFunctions in an Angular Domain

TheAhlfors-Shimizu characteristic is important in this paperLet us recall its definition Suppose that 119864 is a nonempty sub-set of C we denote

119878 (119903 119864 119891) =1

120587intint119864

(

100381610038161003816100381610038161198911015840(119911)

10038161003816100381610038161003816

1 +1003816100381610038161003816119891(119911)

10038161003816100381610038162)

2

119889119908

1198790 (119903 119864 119891) = int119903

0

119878 (119905 119864 119891)

119905119889119905

(13)

The Scientific World Journal 3

When 119864 = C we write 119879(119903C 119891) by 1198790(119903 119891) Then fromTheorem 14 in [6] we have

1003816100381610038161003816119879 (119903 119891) minus log+ 1003816100381610038161003816119891 (0)1003816100381610038161003816 minus 1198790 (119903 119891)

1003816100381610038161003816 le1

2log 2 (14)

And the difference 119879(119903 119891)minus1198790(119903 119891) is a bounded function of119903 so that both the characteristic function1198790(119903 119891) and119879(119903 119891)are interchangeable Denote the following angular domain by

Ω (120579 120576) = 119911 isin C1003816100381610038161003816arg 119911 minus 120579

1003816100381610038161003816 lt 120576 (15)

When 119864 is a sector 119911 isin C |119911| lt 119903 cap Ω(120579 120576) we denote119878(119864 119891) = 119878(119903 Ω(120579 120576) 119891) and

119879 (119903 Ω (120579 120576) 119891) = int119903

0

119878 (119905 Ω (120579 120576) 119891)

119905119889119905 (16)

For any 119886 isin Cinfin and 119886 =infin let 119899(119903 120579 120576 119886) be the number ofzeros counted according to their multiplicities of 119891(119911)minus119886 inthe sector 119911 isin C |119911| lt 119903capΩ(120579 120576) and let 119899119897)(119903 120579 120576 119886) be thenumber of zeros with multiplicities le 119897 of 119891(119911)minus119886 in the sec-tor 119911 isin C |119911| lt 119903 cap Ω(120579 120576) where 119897 is any positive integerSimilarly note the number of poles of 119891 by 119899(119903 120579 120576infin) and119899119897)(119903 120579 120576infin) Denote

119873(119903 120579 120576 119886) = int119903

0

119899 (119905 120579 120576 119886) minus 119899 (0 120579 120576 119886)

119905119889119905

+ 119899 (0 120579 120576 119886) log 119903

119873119897) (119903 120579 120576 119886) = int119903

0

119899119897) (119905 120579 120576 119886) minus 119899119897) (0 120579 120576 119886)

119905119889119905

+ 119899119897) (0 120579 120576 119886) log 119903

(17)

In addition we also need the notations (see [7])

119871 (119903 1205951 1205952) = int1205952

1205951

100381610038161003816100381610038161198911015840(119903119890119894120595)

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119903119890119894120595)10038161003816100381610038162)119903 119889120595

119871 (119903 120595) = int119903

1

100381610038161003816100381610038161198911015840(119905119890119894120595)

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894120595)10038161003816100381610038162)119889119905

(18)

In this section we will establish a fundamental inequalityfor meromorphic functions in an angular domain Firstly wegive the following lemma

Lemma 4 Suppose that 119891(119911) is a meromorphic function and119897V (V = 1 2 119902) be 119902 positive integers and 119886V are 119902(gt 2)distinct points on 119870 and without a pair of 119886V such that theirspherical distance is less than 120575 + 21205753 119899119897V)V be the number ofzeros of119891(119911)minus119886V which are consisted of notmore than 119897V multi-plicities then

119902

sumV=1

(119897V

119897V + 1) 119899119897V)V ge (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 minus

119862 + 9119902ℎ

1205753119871

(19)

Proof Let119863V be a spherical diskwith the center 119886V with radius1205753 on119870 By Theorem 3 we have

119902

sumV=1

(119897V

119897V + 1) 119899119897V)V ge (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 minus

119862 + 9119902ℎ

1205753119871

(20)

Note that 119899119897V)V (119863V) le 119899119897V)V (119886V) whenever 119886V in the island of 119863Vor in the peninsula of119863V Therefore Lemma 4 follows

We are now in the position to establish the main result inthis section

Theorem 5 Let 119891(119911) be a meromorphic function and 119897V (V =1 2 119902) 119902 positive integers If 119886V are 119902 distinct points on119870then one has

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V)

+21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log 119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891)

+ 119867119871 (1 120579 minus 120575 120579 + 120575) + 119867119871 (119903 120579 minus 120575 120579 + 120575)

(21)

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log2119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (1Ω (120579 120593) 119891)

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891) log 119903

+ 119867119871 (1 120579 minus 120575 120579 + 120575) log 119903 + 120594 (119903 120579 minus 120575 120579 + 120575)(22)

for any 120593 0 lt 120593 lt 120575 where119867 is a constant depending only on119886V V = 1 2 119902 and 120594(119903 120579 minus 120575 120579 + 120575) = 119867int

119903

1(119871(119905 120579 minus 120575 120579 +

120575)119905)119889119905

4 The Scientific World Journal

Proof Put 119863119903 = 119911 isin C 1 lt |119911| lt 119903 cap Ω(120579 120593) and 1198650 = 119870 minus119886V Using Lemma 4 we have

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2) [119878 (119903 Ω (120579 120593) 119891)

minus119878 (1 Ω (120579 120593) 119891)]

le

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V) + 119867119871 (119903)

(23)

where119867 = (119862 + 9119902ℎ)1198971205753 which depends only on 1198650 that isonly on 119886V V = 1 2 119902 and

119871 (119903) = 119871 (119903 120579 minus 120593 120579 + 120593) + 119871 (1 120579 minus 120593 120579 + 120593)

+ 119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)

le 119871 (119903 120579 minus 120575 120579 + 120575) + 119871 (1 120579 minus 120575 120579 + 120575)

+ 119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)

(24)

Hence

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)

times [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

minus

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V) minus 119867119871 (119903 120579 minus 120575 120579 + 120575)

minus 119867119871 (1 120579 minus 120575 120579 + 120575)

le 119867 [119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]

(25)

Denote the left expression of (25) by 119860(119903 120593) thus

119889 (119860 (119903 120593))

119889120593

= (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)

times119889 [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

119889120593

(26)

We claim the fact that

[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]2le

2120587

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

times119889 (119860 (119903 120593))

119889120593log 119903

(27)

In fact it follows from the definition of 119871(119903 120595) and Schwarzrsquosinequality that

[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]2

le 2[

[

(int119903

1

100381610038161003816100381610038161198911015840(119905119890119894(120579minus120593))

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894(120579minus120593))

10038161003816100381610038162)119889119905)

2

+ (int119903

1

100381610038161003816100381610038161198911015840(119905119890119894(120579+120593))

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894(120579+120593))

10038161003816100381610038162)119889119905)

2

]

]

le 2120587119889 [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

119889120593log 119903

=2120587

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

119889 (119860 (119903 120593))

119889120593log 119903

(28)

Noting119860(119903 120593) is an increasing function of 120593 we see that thenthere exists a 1205750 gt 0 such that 119860(119903 120593) gt 0 when 120593 gt 1205750 and119860(119903 120593) le 0 when 120593 le 1205750 For 120593 gt 1205750 by (25) and (27)

[119860 (119903 120593)]2le 1198672[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]

2

le21205871198672

(sum119902

119895=1(1 minus

1

119897V + 1) minus 2)

log 119903119889 (119860 (119903 120593))

119889120593

(29)

that is

119889120593 le21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

log 119903119889 (119860 (119903 120593))

[119860 (119903 120593)]2 (30)

Integrating each side of the inequality leads to

120575 minus 120593 = int120575

120593

119889120593 le21205871198672

(sum119902

119895=1(1 minus

1

119897V + 1) minus 2)119860 (119903 120593)

log 119903

(31)

Thus

119860 (119903 120593) le21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120591)

log 119903 (32)

On the case of 120593 le 1205750 the above inequality is obviously validbecause of 119860(119903 120593) le 0 Replacing 119860(119903 120593) in the above

The Scientific World Journal 5

inequality with its explicit expression we see that (21) is esta-blished Therefore

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+1205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log2119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (1Ω (120579 120593) 119891)

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891) log 119903

+ 119867119871 (1 120579 minus 120575 120579 + 120575) log 119903 + 120594 (119903 120579 minus 120575 120579 + 120575) (33)

where 120594(119903 120579 minus 120575 120579 + 120575) = 119867int119903

1(119871(119905 120579 minus 120575 120579 + 120575)119905)119889119905

Lemma 6 (Zhang [7]) Under the condition ofTheorem 5 onehas

120594 (119903 120579 minus 120575 120579 + 120575) = 119867int119903

1

119871 (119905 120579 minus 120575 120579 + 120575)

119905119889119905

le 119867radic2120575120587119878 (119903 Ω (120579 120575) 119891) log 119903(34)

or

120594 (119903 120579 minus 120575 120579 + 120575) le 119867radic2120575120587119879 (119903 Ω (120579 120575) 119891)

times log119879 (119903 Ω (120579 120575) 119891)

(35)

with at most one exceptional set 119864120575 of 119903 where 119864120575 consists of aseries of intervals and satisfies

int119864120575

1

119903 log 119903119889119903 le

1

log119879 (119903 Ω (120579 120575) 119891)lt infin (36)

In particular if the order of 119891(119911) is 120588 (0 lt 120588 lt +infin) then

120594 (119903 120579 minus 120575 120579 + 120575) le 119874 (11990331205884) (37)

FromTheorem 3 and Lemma 6 we can write the result inTheorem 3 as

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+ 119874 (log2119903) + 120594 (119903 120579 minus 120575 120579 + 120575)

(38)

If the order of119891(119911) is120588 (0 lt 120588 lt +infin) then the inequalitywillbe

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V) + 119874 (11990331205884)

(39)

4 Bloch Direction of Meromorphic Functions

In this section we will research the singular direction corre-sponding toTheoremA Suppose that119891(119911) is a meromorphicfunction of infinite order Then there is a real function 120588(119903)called anHiongrsquos proximate order (see [8]) of119891(119911) which hasthe following properties (i) 120588(119903) is continuous and nonde-creasing for 119903 ge 1199030 (1199030 gt 0) and tends to +infin as 119903 rarr +infin (ii)The function 119880(119903) = 119903120588(119903)(119903 ge 1199030) satisfies the condition

lim119903rarr+infin

log119880 (119877)

log119880 (119903)= 1 119877 = 119903 +

119903

log119880 (119903)

lim sup119903rarr+infin

log119879 (119903 119891)log119880 (119903)

= 1

(40)

For a meromorphic function of infinite order ZhuangQitai (or Chuang Chitai) [9] gives the following definition ofBorel direction and Bloch direction

Definition 7 Let 119891(119911) be a meromorphic function of infiniteorder and 120588(119903) an order of119891(119911) A direction arg 119911 = 120579 is calleda Borel direction of order 120588(119903) of 119891(119911) if no matter how smallthe positive number 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)120588 (119903) log 119903

= 1 (41)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588(119903) of119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that119896119895 is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (42)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)120588 (119903) log 119903

= 1 (43)

For the connection of Borel direction and Bloch directionof meromorphic function of infinite order Chuang [9] hasproved the following theorem

Theorem C Let 119891(119911) be a meromorphic function of infiniteorder and 120588(119903) an order of 119891(119911) Then every Borel direction oforder 120588(119903) of 119891(119911) is a Bloch direction of order 120588(119903) of 119891(119911)

6 The Scientific World Journal

It is natural to consider whether there exists a similarresult if meromorphic function of order infinity is replacedwithmeromorphic function of order 120588 (0 lt 120588 lt +infin) In thissection we extend the above theorem to meromorphic func-tion of order 120588 (0 lt 120588 lt +infin)

Definition 8 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) A direction arg 119911 = 120579 is called a Borel dir-ection of order 120588 of 119891(119911) if no matter how small the positivenumber 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)log 119903

= 120588 (44)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588 of 119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (45)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)log 119903

= 120588 (46)

Theorem 9 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then every Borel direction of order 120588 of 119891(119911)is a Bloch direction of order 120588 of 119891(119911)

In order to prove Theorem 9 we need the followinglemma

Lemma 10 (Zhang [7]) Let 119891(119911) be a meromorphic functionof order 120588 (0 lt 120588 lt +infin) Then a direction arg 119911 = 120579 is a Boreldirection of order 120588 of 119891(119911) if and only if it satisfies

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (47)

for any 120576 (0 lt 120576 lt 1205872)

We are now in the position to proveTheorem 9

Proof Suppose that arg 119911 = 120579 is a Borel direction of order 120588 of119891(119911) then for any 120576 (0 lt 120576 lt 1205872) we have

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (48)

If arg 119911 = 120579 is not a Bloch direction of order 120588 of 119891(119911) thenthere exit a system 119886119895 (119895 = 1 2 119902) of distinct values and asystem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (49)

And for any integer 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (50)

Hence we can get

lim sup119903rarrinfin

log119873119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (51)

for any integer 119895 (1 le 119895 le 119902)Therefore we can find a positivenumber 120591 lt 120588 such that

119873119896119895) (119903 120579 2120576 119886119895) le 119903120591 (52)

By (39) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Ω (120579 120576) 119891)

le

119902

sumV=1

119896119895

119896119895 + 1119873119896119895) (119903 120579 2120576 119886V) + 119874 (11990331205884)

le 119874 (119903120577)

(53)

where 120577 = max120591 31205884 lt 120588Hence

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120577 lt 120588 (54)

This contradicts with (48) andTheorem 9 follows

Corollary 11 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then there is a direction arg 119911 = 120579 which is aBloch direction of order 120588 of 119891(119911)

Note that Corollary 11 is a corresponding result of Theo-rem A in angular distribution

5 Pseudo-T Direction ofMeromorphic Functions

In 2003 Zheng [1] introduced a new singular direction calledT direction We call 119869 arg 119911 = 120579 the T direction of 119891(119911)provided that given any 119886 isin Cinfin possiblywith exception of atmost two values of 119886 for any positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873(119903 120579 120576 119886)

119879 (119903 119891)gt 0 (55)

For the existence of T direction of meromorphic function119891(119911) Guo et al [10] proved the followingTheorem

Theorem C Let 119891(119911) be a meromorphic function and satisfy

lim sup119903rarrinfin

119879 (119903 119891)

log2119903= infin (56)

Then 119891(119911)must have a T direction

The Scientific World Journal 7

Theorem C was conjectured by Zheng [1] In [11] theauthors study the existence of T direction of 119891(119911) concerningmultiple values We call 119869 arg 119911 = 120579 the T direction of 119891(119911)concerningmultiple values provided that given any 119886 isin Cinfinpossibly with exception of at most [(2119897 + 2)119897] values of 119886 forany positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873119897) (119903 120579 120576 119886)

119879 (119903 119891)gt 0 (57)

where [119909] implies the maximum integer number which doesnot exceed 119909 and 119897 is a positive integer

Theorem D Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a T direction of 119891(119911) concerningmultiple values

Note that the T direction of meromorphic function con-cerning multiple values is a refinement of the ordinary T dir-ection since [(2119897+2)119897] rarr 2 as 119897 rarr infin Since Zheng [1] gavethe definition of T direction then there is a considerablenumber result related this directionwe refer the reader to [12]for finding a careful discussion of this direction

It is well known that T direction is a concept in angulardistribution which corresponds to the Nevanlinna secondfundamental theorem in module distribution It is natural toconsider the corresponding result to Theorem B in angulardistribution

Definition 12 Let 119891(119911) be a meromorphic function A direc-tion arg 119911 = 120579 is called a pseudo-T direction of119891(119911) if for anynumber 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (58)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 120579 120576 119886119895)

119879 (119903 119891)gt 0 (59)

Theorem 13 Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a pseudo-T direction of 119891(119911)

Remark 14 (i) In Theorem C 119902 = 3 119896119895 = infin (119895 = 1 2 3) soTheorem C is a special case of Theorem 13

(ii) If 119896119895 = 1 (119895 = 1 2 119902) then 119902 = 5 if 119896119895 = 2 (119895 =1 2 119902) then 119902 = 4 if 119896119895 = 119897 ge 3 (119895 = 1 2 119902) then119902 = 3 So Theorem D is a special case of Theorem 13

In order to prove Theorem 13 we need the followinglemma

Lemma 15 (Li and Gu [13] see also Xuan [14]) Suppose thatΨ(119903) is a nonnegative increasing function in (1infin) and satisfies

lim sup119903rarrinfin

Ψ (119903)

log2119903= infin (60)

Then for any set 119864 sub (1infin) such that int119864(1119903 log 119903)119889119903 lt 13

one has

lim sup119903rarrinfin119903isin(1infin)minus119864

Ψ (119903)

log2119903= infin (61)

Proof Firstly we prove the following statement Let 119898 (119898 ge4) be a fixed positive integer 1205790 = 0 1205791 = 2120587119898 120579119898minus1 =(119898 minus 1)2120587119898 120579119898 = 1205790 We put Δ(120579119894) = 119911 | arg 119911 minus 120579119894| lt2120587119898 Δ119900(120579119894) = 119911 | arg 119911 minus 120579119894| lt 120587119898 119894 = 0 1 119898 minus 1Δ(120579119898) = Δ(1205790) Δ

119900(120579119898) = Δ119900(1205790) Then among these119898 angu-lar domains Δ(120579119894) there is at least an angular domain Δ(120579119894)such that for any system 119886119895 (119895 = 1 2 119902) of distinct valuesand any system 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positiveinteger or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (62)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (63)

Otherwise for any angular domainΔ(120579119894) (1 le 119894 le 119898) there isa system 119886

119895

119894(119895 = 1 2 119902) of distinct values and a system

119896119895

119894(119895 = 1 2 119902) such that 119896119895

119894is a positive integer or+infin and

that

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

) gt 2 (64)

for any 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

119873119896119895

119894) (119903 Δ (120579119894) 119886

119895

119894)

119879 (119903 119891)= 0 (65)

Put

119902

sum119895=1

(1 minus1

119896119895 + 1) = min1le119894le119898

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

)

gt 2 (66)

ApplyingTheorem 5 to Δ119900(120579119894+1) Δ(120579119894+1) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Δ119900 (120579119894+1) 119891)

le

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895

119894) (119903 Δ (120579119894+1) 119886

119895

119894+1)

+ 119874 (log2119903) + 120594 (119903 Δ (120579119894+1))

(67)

8 The Scientific World Journal

Noting119879(119903 119891) = sum119898minus1

119894=0119879(119903 Δ119900(120579119894+1) 119891) and adding two sides

of the above expression from 119894 = 0 to119898 minus 1 we can obtain

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 119891)

le119898minus1

sum119894=0

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895) (119903 Δ (120579119894+1) 119886119895

119894+1)

+ 119874 (log2119903) +119898minus1

sum119894=0

120594 (119903 Δ (120579119894+1))

(68)

For any 119894 there exists a 119903119894 the inequality 119879(119903 Δ119900(120579119894+1) 119891) gt

1198903119898 would bold for 119903 gt 119903119894 while the inequality (22) does notlook appropriate here Put 119864Δ119900(120579119894+1) is the set of 119903 which con-sists of a series of intervals and satisfies

int119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt

1

3119898 (69)

Let 1199030 = max119903119894 119894 = 1 2 119898 we have for any 119894 119879(1199030Δ119900(120579119894+1) 119891) gt 1198903119898 then

intcup119898minus1119894=0119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le119898minus1

sum119894=0

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt1

3

(70)

Applying Lemma 15 we have

lim sup119903rarrinfin119903isin(1infin)minus119864

119879 (119903 119891)

log2119903= infin (71)

where 119864 = cup119898minus1119894=0

119864Δ119900(120579119894+1) Therefore there exists a sequence1199031015840119899isin (1infin) minus 119864 such that

lim119899rarrinfin

119879 (1199031015840119899 119891)

log21199031015840119899

= infin (72)

It follows from (38) (68) and (72) that

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2) le 0 (73)

Hence119902

sum119895=1

(1 minus1

119896119895 + 1) le 2 (74)

This is a contradiction Hence for an arbitrary positive inte-ger 119898 there is at least an angular domain Δ(120579119894) such that forany system 119886119895 (119895 = 1 2 119902) of distinct values and any sys-tem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (75)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (76)

Choosing subsequence of 120579119898 still denote it 120579119898 we assumethat 120579119898 rarr 1205790 Put 119871 arg 119911 = 1205790 then 119871 is a pseudo-Tdirection that is stated in Definition 12

In fact for any 120576 (0 lt 120576 lt 1205872) when 119898 is sufficientlylarge we have Δ(120579119898) sub Ω(1205790 120576) By (76) we have

lim sup119903rarrinfin

119873119896119895) (119903 1205790 120576 119886119895)

119879 (119903 119891)ge lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119898) 119886119895)

119879 (119903 119891)gt 0

(77)

Hence Theorem 13 holds in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The first author was partly supported by the National NaturalScience Foundation of China (Grant no 11201395) and by theScience Foundation of Educational Commission of HubeiProvince (Grant no Q20132801) The second author waspartly supported by the NSF of Jiangxi Province (Grant20122BAB201006) The third author was partly supported byBeijing Natural Science Foundation (Grant no 1132013)

References

[1] J H Zheng ldquoOn transcendental meromorphic functions withradially distributed valuesrdquo Science in China A vol 47 no 3 pp401ndash416 2004

[2] L Yang Value Distribution Theory Springer Berlin Germany1993

[3] M Tsuji PotentialTheory inModern FunctionTheoryMaruzenTokyo Japan 1959

[4] D C Sun ldquoMain theorem on covering surfacesrdquo Acta Mathe-matica Scientia B vol 14 no 2 pp 213ndash225 1994

[5] D C Sun ldquoInequalities for covering surfaces and their applica-tionsrdquoChinese Annals ofMathematics A vol 18 no 1 pp 59ndash641997 (Chinese)

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs Clarendon Press Oxford UK 1964

[7] X L Zhang ldquoA fundamental inequality for meromorphic func-tions in an angular domain and its applicationrdquoActaMathemat-ica Sinica vol 10 no 3 pp 308ndash314 1994

[8] Q T Zhuang Singular Directions of Meromorphic Functionsvol 11 of Series of Monographs in Pure and AppliedMathematicsScience Press Beijing China 1982 (Chinese)

[9] C T Chuang ldquoOn Borel directions of meromorphic functionsof infinite order IIrdquoBulletin of theHongKongMathematical Soc-iety vol 2 no 2 pp 305ndash323 1999

[10] H Guo J H Zheng and TW Ng ldquoOn a new singular directionof meromorphic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 69 no 2 pp 277ndash287 2004

The Scientific World Journal 9

[11] Z-J Wu and D-C Sun ldquoOn the existence of 119879 direction ofmeromorphic function concerning multiple valuesrdquo KodaiMathematical Journal vol 31 no 1 pp 133ndash149 2008

[12] J H Zheng Value Distribution of Meromorphic FunctionsTsinghua University Press Beijing China 2010

[13] C H Li and Y X Gu ldquoA fundamental inequality for 119870-quasimeromorphic mappings in an angular domain and itsapplicationrdquo Acta Mathematica Sinica vol 49 no 6 pp 1279ndash1286 2006 (Chinese)

[14] Z-X Xuan ldquoOn the existence of 119879-direction of algebroidfunctions a problem of J H Zhengrdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 540ndash547 2008

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

When 119864 = C we write 119879(119903C 119891) by 1198790(119903 119891) Then fromTheorem 14 in [6] we have

1003816100381610038161003816119879 (119903 119891) minus log+ 1003816100381610038161003816119891 (0)1003816100381610038161003816 minus 1198790 (119903 119891)

1003816100381610038161003816 le1

2log 2 (14)

And the difference 119879(119903 119891)minus1198790(119903 119891) is a bounded function of119903 so that both the characteristic function1198790(119903 119891) and119879(119903 119891)are interchangeable Denote the following angular domain by

Ω (120579 120576) = 119911 isin C1003816100381610038161003816arg 119911 minus 120579

1003816100381610038161003816 lt 120576 (15)

When 119864 is a sector 119911 isin C |119911| lt 119903 cap Ω(120579 120576) we denote119878(119864 119891) = 119878(119903 Ω(120579 120576) 119891) and

119879 (119903 Ω (120579 120576) 119891) = int119903

0

119878 (119905 Ω (120579 120576) 119891)

119905119889119905 (16)

For any 119886 isin Cinfin and 119886 =infin let 119899(119903 120579 120576 119886) be the number ofzeros counted according to their multiplicities of 119891(119911)minus119886 inthe sector 119911 isin C |119911| lt 119903capΩ(120579 120576) and let 119899119897)(119903 120579 120576 119886) be thenumber of zeros with multiplicities le 119897 of 119891(119911)minus119886 in the sec-tor 119911 isin C |119911| lt 119903 cap Ω(120579 120576) where 119897 is any positive integerSimilarly note the number of poles of 119891 by 119899(119903 120579 120576infin) and119899119897)(119903 120579 120576infin) Denote

119873(119903 120579 120576 119886) = int119903

0

119899 (119905 120579 120576 119886) minus 119899 (0 120579 120576 119886)

119905119889119905

+ 119899 (0 120579 120576 119886) log 119903

119873119897) (119903 120579 120576 119886) = int119903

0

119899119897) (119905 120579 120576 119886) minus 119899119897) (0 120579 120576 119886)

119905119889119905

+ 119899119897) (0 120579 120576 119886) log 119903

(17)

In addition we also need the notations (see [7])

119871 (119903 1205951 1205952) = int1205952

1205951

100381610038161003816100381610038161198911015840(119903119890119894120595)

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119903119890119894120595)10038161003816100381610038162)119903 119889120595

119871 (119903 120595) = int119903

1

100381610038161003816100381610038161198911015840(119905119890119894120595)

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894120595)10038161003816100381610038162)119889119905

(18)

In this section we will establish a fundamental inequalityfor meromorphic functions in an angular domain Firstly wegive the following lemma

Lemma 4 Suppose that 119891(119911) is a meromorphic function and119897V (V = 1 2 119902) be 119902 positive integers and 119886V are 119902(gt 2)distinct points on 119870 and without a pair of 119886V such that theirspherical distance is less than 120575 + 21205753 119899119897V)V be the number ofzeros of119891(119911)minus119886V which are consisted of notmore than 119897V multi-plicities then

119902

sumV=1

(119897V

119897V + 1) 119899119897V)V ge (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 minus

119862 + 9119902ℎ

1205753119871

(19)

Proof Let119863V be a spherical diskwith the center 119886V with radius1205753 on119870 By Theorem 3 we have

119902

sumV=1

(119897V

119897V + 1) 119899119897V)V ge (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 minus

119862 + 9119902ℎ

1205753119871

(20)

Note that 119899119897V)V (119863V) le 119899119897V)V (119886V) whenever 119886V in the island of 119863Vor in the peninsula of119863V Therefore Lemma 4 follows

We are now in the position to establish the main result inthis section

Theorem 5 Let 119891(119911) be a meromorphic function and 119897V (V =1 2 119902) 119902 positive integers If 119886V are 119902 distinct points on119870then one has

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V)

+21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log 119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891)

+ 119867119871 (1 120579 minus 120575 120579 + 120575) + 119867119871 (119903 120579 minus 120575 120579 + 120575)

(21)

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log2119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (1Ω (120579 120593) 119891)

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891) log 119903

+ 119867119871 (1 120579 minus 120575 120579 + 120575) log 119903 + 120594 (119903 120579 minus 120575 120579 + 120575)(22)

for any 120593 0 lt 120593 lt 120575 where119867 is a constant depending only on119886V V = 1 2 119902 and 120594(119903 120579 minus 120575 120579 + 120575) = 119867int

119903

1(119871(119905 120579 minus 120575 120579 +

120575)119905)119889119905

4 The Scientific World Journal

Proof Put 119863119903 = 119911 isin C 1 lt |119911| lt 119903 cap Ω(120579 120593) and 1198650 = 119870 minus119886V Using Lemma 4 we have

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2) [119878 (119903 Ω (120579 120593) 119891)

minus119878 (1 Ω (120579 120593) 119891)]

le

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V) + 119867119871 (119903)

(23)

where119867 = (119862 + 9119902ℎ)1198971205753 which depends only on 1198650 that isonly on 119886V V = 1 2 119902 and

119871 (119903) = 119871 (119903 120579 minus 120593 120579 + 120593) + 119871 (1 120579 minus 120593 120579 + 120593)

+ 119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)

le 119871 (119903 120579 minus 120575 120579 + 120575) + 119871 (1 120579 minus 120575 120579 + 120575)

+ 119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)

(24)

Hence

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)

times [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

minus

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V) minus 119867119871 (119903 120579 minus 120575 120579 + 120575)

minus 119867119871 (1 120579 minus 120575 120579 + 120575)

le 119867 [119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]

(25)

Denote the left expression of (25) by 119860(119903 120593) thus

119889 (119860 (119903 120593))

119889120593

= (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)

times119889 [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

119889120593

(26)

We claim the fact that

[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]2le

2120587

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

times119889 (119860 (119903 120593))

119889120593log 119903

(27)

In fact it follows from the definition of 119871(119903 120595) and Schwarzrsquosinequality that

[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]2

le 2[

[

(int119903

1

100381610038161003816100381610038161198911015840(119905119890119894(120579minus120593))

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894(120579minus120593))

10038161003816100381610038162)119889119905)

2

+ (int119903

1

100381610038161003816100381610038161198911015840(119905119890119894(120579+120593))

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894(120579+120593))

10038161003816100381610038162)119889119905)

2

]

]

le 2120587119889 [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

119889120593log 119903

=2120587

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

119889 (119860 (119903 120593))

119889120593log 119903

(28)

Noting119860(119903 120593) is an increasing function of 120593 we see that thenthere exists a 1205750 gt 0 such that 119860(119903 120593) gt 0 when 120593 gt 1205750 and119860(119903 120593) le 0 when 120593 le 1205750 For 120593 gt 1205750 by (25) and (27)

[119860 (119903 120593)]2le 1198672[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]

2

le21205871198672

(sum119902

119895=1(1 minus

1

119897V + 1) minus 2)

log 119903119889 (119860 (119903 120593))

119889120593

(29)

that is

119889120593 le21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

log 119903119889 (119860 (119903 120593))

[119860 (119903 120593)]2 (30)

Integrating each side of the inequality leads to

120575 minus 120593 = int120575

120593

119889120593 le21205871198672

(sum119902

119895=1(1 minus

1

119897V + 1) minus 2)119860 (119903 120593)

log 119903

(31)

Thus

119860 (119903 120593) le21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120591)

log 119903 (32)

On the case of 120593 le 1205750 the above inequality is obviously validbecause of 119860(119903 120593) le 0 Replacing 119860(119903 120593) in the above

The Scientific World Journal 5

inequality with its explicit expression we see that (21) is esta-blished Therefore

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+1205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log2119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (1Ω (120579 120593) 119891)

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891) log 119903

+ 119867119871 (1 120579 minus 120575 120579 + 120575) log 119903 + 120594 (119903 120579 minus 120575 120579 + 120575) (33)

where 120594(119903 120579 minus 120575 120579 + 120575) = 119867int119903

1(119871(119905 120579 minus 120575 120579 + 120575)119905)119889119905

Lemma 6 (Zhang [7]) Under the condition ofTheorem 5 onehas

120594 (119903 120579 minus 120575 120579 + 120575) = 119867int119903

1

119871 (119905 120579 minus 120575 120579 + 120575)

119905119889119905

le 119867radic2120575120587119878 (119903 Ω (120579 120575) 119891) log 119903(34)

or

120594 (119903 120579 minus 120575 120579 + 120575) le 119867radic2120575120587119879 (119903 Ω (120579 120575) 119891)

times log119879 (119903 Ω (120579 120575) 119891)

(35)

with at most one exceptional set 119864120575 of 119903 where 119864120575 consists of aseries of intervals and satisfies

int119864120575

1

119903 log 119903119889119903 le

1

log119879 (119903 Ω (120579 120575) 119891)lt infin (36)

In particular if the order of 119891(119911) is 120588 (0 lt 120588 lt +infin) then

120594 (119903 120579 minus 120575 120579 + 120575) le 119874 (11990331205884) (37)

FromTheorem 3 and Lemma 6 we can write the result inTheorem 3 as

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+ 119874 (log2119903) + 120594 (119903 120579 minus 120575 120579 + 120575)

(38)

If the order of119891(119911) is120588 (0 lt 120588 lt +infin) then the inequalitywillbe

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V) + 119874 (11990331205884)

(39)

4 Bloch Direction of Meromorphic Functions

In this section we will research the singular direction corre-sponding toTheoremA Suppose that119891(119911) is a meromorphicfunction of infinite order Then there is a real function 120588(119903)called anHiongrsquos proximate order (see [8]) of119891(119911) which hasthe following properties (i) 120588(119903) is continuous and nonde-creasing for 119903 ge 1199030 (1199030 gt 0) and tends to +infin as 119903 rarr +infin (ii)The function 119880(119903) = 119903120588(119903)(119903 ge 1199030) satisfies the condition

lim119903rarr+infin

log119880 (119877)

log119880 (119903)= 1 119877 = 119903 +

119903

log119880 (119903)

lim sup119903rarr+infin

log119879 (119903 119891)log119880 (119903)

= 1

(40)

For a meromorphic function of infinite order ZhuangQitai (or Chuang Chitai) [9] gives the following definition ofBorel direction and Bloch direction

Definition 7 Let 119891(119911) be a meromorphic function of infiniteorder and 120588(119903) an order of119891(119911) A direction arg 119911 = 120579 is calleda Borel direction of order 120588(119903) of 119891(119911) if no matter how smallthe positive number 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)120588 (119903) log 119903

= 1 (41)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588(119903) of119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that119896119895 is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (42)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)120588 (119903) log 119903

= 1 (43)

For the connection of Borel direction and Bloch directionof meromorphic function of infinite order Chuang [9] hasproved the following theorem

Theorem C Let 119891(119911) be a meromorphic function of infiniteorder and 120588(119903) an order of 119891(119911) Then every Borel direction oforder 120588(119903) of 119891(119911) is a Bloch direction of order 120588(119903) of 119891(119911)

6 The Scientific World Journal

It is natural to consider whether there exists a similarresult if meromorphic function of order infinity is replacedwithmeromorphic function of order 120588 (0 lt 120588 lt +infin) In thissection we extend the above theorem to meromorphic func-tion of order 120588 (0 lt 120588 lt +infin)

Definition 8 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) A direction arg 119911 = 120579 is called a Borel dir-ection of order 120588 of 119891(119911) if no matter how small the positivenumber 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)log 119903

= 120588 (44)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588 of 119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (45)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)log 119903

= 120588 (46)

Theorem 9 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then every Borel direction of order 120588 of 119891(119911)is a Bloch direction of order 120588 of 119891(119911)

In order to prove Theorem 9 we need the followinglemma

Lemma 10 (Zhang [7]) Let 119891(119911) be a meromorphic functionof order 120588 (0 lt 120588 lt +infin) Then a direction arg 119911 = 120579 is a Boreldirection of order 120588 of 119891(119911) if and only if it satisfies

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (47)

for any 120576 (0 lt 120576 lt 1205872)

We are now in the position to proveTheorem 9

Proof Suppose that arg 119911 = 120579 is a Borel direction of order 120588 of119891(119911) then for any 120576 (0 lt 120576 lt 1205872) we have

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (48)

If arg 119911 = 120579 is not a Bloch direction of order 120588 of 119891(119911) thenthere exit a system 119886119895 (119895 = 1 2 119902) of distinct values and asystem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (49)

And for any integer 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (50)

Hence we can get

lim sup119903rarrinfin

log119873119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (51)

for any integer 119895 (1 le 119895 le 119902)Therefore we can find a positivenumber 120591 lt 120588 such that

119873119896119895) (119903 120579 2120576 119886119895) le 119903120591 (52)

By (39) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Ω (120579 120576) 119891)

le

119902

sumV=1

119896119895

119896119895 + 1119873119896119895) (119903 120579 2120576 119886V) + 119874 (11990331205884)

le 119874 (119903120577)

(53)

where 120577 = max120591 31205884 lt 120588Hence

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120577 lt 120588 (54)

This contradicts with (48) andTheorem 9 follows

Corollary 11 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then there is a direction arg 119911 = 120579 which is aBloch direction of order 120588 of 119891(119911)

Note that Corollary 11 is a corresponding result of Theo-rem A in angular distribution

5 Pseudo-T Direction ofMeromorphic Functions

In 2003 Zheng [1] introduced a new singular direction calledT direction We call 119869 arg 119911 = 120579 the T direction of 119891(119911)provided that given any 119886 isin Cinfin possiblywith exception of atmost two values of 119886 for any positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873(119903 120579 120576 119886)

119879 (119903 119891)gt 0 (55)

For the existence of T direction of meromorphic function119891(119911) Guo et al [10] proved the followingTheorem

Theorem C Let 119891(119911) be a meromorphic function and satisfy

lim sup119903rarrinfin

119879 (119903 119891)

log2119903= infin (56)

Then 119891(119911)must have a T direction

The Scientific World Journal 7

Theorem C was conjectured by Zheng [1] In [11] theauthors study the existence of T direction of 119891(119911) concerningmultiple values We call 119869 arg 119911 = 120579 the T direction of 119891(119911)concerningmultiple values provided that given any 119886 isin Cinfinpossibly with exception of at most [(2119897 + 2)119897] values of 119886 forany positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873119897) (119903 120579 120576 119886)

119879 (119903 119891)gt 0 (57)

where [119909] implies the maximum integer number which doesnot exceed 119909 and 119897 is a positive integer

Theorem D Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a T direction of 119891(119911) concerningmultiple values

Note that the T direction of meromorphic function con-cerning multiple values is a refinement of the ordinary T dir-ection since [(2119897+2)119897] rarr 2 as 119897 rarr infin Since Zheng [1] gavethe definition of T direction then there is a considerablenumber result related this directionwe refer the reader to [12]for finding a careful discussion of this direction

It is well known that T direction is a concept in angulardistribution which corresponds to the Nevanlinna secondfundamental theorem in module distribution It is natural toconsider the corresponding result to Theorem B in angulardistribution

Definition 12 Let 119891(119911) be a meromorphic function A direc-tion arg 119911 = 120579 is called a pseudo-T direction of119891(119911) if for anynumber 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (58)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 120579 120576 119886119895)

119879 (119903 119891)gt 0 (59)

Theorem 13 Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a pseudo-T direction of 119891(119911)

Remark 14 (i) In Theorem C 119902 = 3 119896119895 = infin (119895 = 1 2 3) soTheorem C is a special case of Theorem 13

(ii) If 119896119895 = 1 (119895 = 1 2 119902) then 119902 = 5 if 119896119895 = 2 (119895 =1 2 119902) then 119902 = 4 if 119896119895 = 119897 ge 3 (119895 = 1 2 119902) then119902 = 3 So Theorem D is a special case of Theorem 13

In order to prove Theorem 13 we need the followinglemma

Lemma 15 (Li and Gu [13] see also Xuan [14]) Suppose thatΨ(119903) is a nonnegative increasing function in (1infin) and satisfies

lim sup119903rarrinfin

Ψ (119903)

log2119903= infin (60)

Then for any set 119864 sub (1infin) such that int119864(1119903 log 119903)119889119903 lt 13

one has

lim sup119903rarrinfin119903isin(1infin)minus119864

Ψ (119903)

log2119903= infin (61)

Proof Firstly we prove the following statement Let 119898 (119898 ge4) be a fixed positive integer 1205790 = 0 1205791 = 2120587119898 120579119898minus1 =(119898 minus 1)2120587119898 120579119898 = 1205790 We put Δ(120579119894) = 119911 | arg 119911 minus 120579119894| lt2120587119898 Δ119900(120579119894) = 119911 | arg 119911 minus 120579119894| lt 120587119898 119894 = 0 1 119898 minus 1Δ(120579119898) = Δ(1205790) Δ

119900(120579119898) = Δ119900(1205790) Then among these119898 angu-lar domains Δ(120579119894) there is at least an angular domain Δ(120579119894)such that for any system 119886119895 (119895 = 1 2 119902) of distinct valuesand any system 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positiveinteger or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (62)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (63)

Otherwise for any angular domainΔ(120579119894) (1 le 119894 le 119898) there isa system 119886

119895

119894(119895 = 1 2 119902) of distinct values and a system

119896119895

119894(119895 = 1 2 119902) such that 119896119895

119894is a positive integer or+infin and

that

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

) gt 2 (64)

for any 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

119873119896119895

119894) (119903 Δ (120579119894) 119886

119895

119894)

119879 (119903 119891)= 0 (65)

Put

119902

sum119895=1

(1 minus1

119896119895 + 1) = min1le119894le119898

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

)

gt 2 (66)

ApplyingTheorem 5 to Δ119900(120579119894+1) Δ(120579119894+1) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Δ119900 (120579119894+1) 119891)

le

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895

119894) (119903 Δ (120579119894+1) 119886

119895

119894+1)

+ 119874 (log2119903) + 120594 (119903 Δ (120579119894+1))

(67)

8 The Scientific World Journal

Noting119879(119903 119891) = sum119898minus1

119894=0119879(119903 Δ119900(120579119894+1) 119891) and adding two sides

of the above expression from 119894 = 0 to119898 minus 1 we can obtain

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 119891)

le119898minus1

sum119894=0

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895) (119903 Δ (120579119894+1) 119886119895

119894+1)

+ 119874 (log2119903) +119898minus1

sum119894=0

120594 (119903 Δ (120579119894+1))

(68)

For any 119894 there exists a 119903119894 the inequality 119879(119903 Δ119900(120579119894+1) 119891) gt

1198903119898 would bold for 119903 gt 119903119894 while the inequality (22) does notlook appropriate here Put 119864Δ119900(120579119894+1) is the set of 119903 which con-sists of a series of intervals and satisfies

int119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt

1

3119898 (69)

Let 1199030 = max119903119894 119894 = 1 2 119898 we have for any 119894 119879(1199030Δ119900(120579119894+1) 119891) gt 1198903119898 then

intcup119898minus1119894=0119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le119898minus1

sum119894=0

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt1

3

(70)

Applying Lemma 15 we have

lim sup119903rarrinfin119903isin(1infin)minus119864

119879 (119903 119891)

log2119903= infin (71)

where 119864 = cup119898minus1119894=0

119864Δ119900(120579119894+1) Therefore there exists a sequence1199031015840119899isin (1infin) minus 119864 such that

lim119899rarrinfin

119879 (1199031015840119899 119891)

log21199031015840119899

= infin (72)

It follows from (38) (68) and (72) that

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2) le 0 (73)

Hence119902

sum119895=1

(1 minus1

119896119895 + 1) le 2 (74)

This is a contradiction Hence for an arbitrary positive inte-ger 119898 there is at least an angular domain Δ(120579119894) such that forany system 119886119895 (119895 = 1 2 119902) of distinct values and any sys-tem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (75)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (76)

Choosing subsequence of 120579119898 still denote it 120579119898 we assumethat 120579119898 rarr 1205790 Put 119871 arg 119911 = 1205790 then 119871 is a pseudo-Tdirection that is stated in Definition 12

In fact for any 120576 (0 lt 120576 lt 1205872) when 119898 is sufficientlylarge we have Δ(120579119898) sub Ω(1205790 120576) By (76) we have

lim sup119903rarrinfin

119873119896119895) (119903 1205790 120576 119886119895)

119879 (119903 119891)ge lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119898) 119886119895)

119879 (119903 119891)gt 0

(77)

Hence Theorem 13 holds in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The first author was partly supported by the National NaturalScience Foundation of China (Grant no 11201395) and by theScience Foundation of Educational Commission of HubeiProvince (Grant no Q20132801) The second author waspartly supported by the NSF of Jiangxi Province (Grant20122BAB201006) The third author was partly supported byBeijing Natural Science Foundation (Grant no 1132013)

References

[1] J H Zheng ldquoOn transcendental meromorphic functions withradially distributed valuesrdquo Science in China A vol 47 no 3 pp401ndash416 2004

[2] L Yang Value Distribution Theory Springer Berlin Germany1993

[3] M Tsuji PotentialTheory inModern FunctionTheoryMaruzenTokyo Japan 1959

[4] D C Sun ldquoMain theorem on covering surfacesrdquo Acta Mathe-matica Scientia B vol 14 no 2 pp 213ndash225 1994

[5] D C Sun ldquoInequalities for covering surfaces and their applica-tionsrdquoChinese Annals ofMathematics A vol 18 no 1 pp 59ndash641997 (Chinese)

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs Clarendon Press Oxford UK 1964

[7] X L Zhang ldquoA fundamental inequality for meromorphic func-tions in an angular domain and its applicationrdquoActaMathemat-ica Sinica vol 10 no 3 pp 308ndash314 1994

[8] Q T Zhuang Singular Directions of Meromorphic Functionsvol 11 of Series of Monographs in Pure and AppliedMathematicsScience Press Beijing China 1982 (Chinese)

[9] C T Chuang ldquoOn Borel directions of meromorphic functionsof infinite order IIrdquoBulletin of theHongKongMathematical Soc-iety vol 2 no 2 pp 305ndash323 1999

[10] H Guo J H Zheng and TW Ng ldquoOn a new singular directionof meromorphic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 69 no 2 pp 277ndash287 2004

The Scientific World Journal 9

[11] Z-J Wu and D-C Sun ldquoOn the existence of 119879 direction ofmeromorphic function concerning multiple valuesrdquo KodaiMathematical Journal vol 31 no 1 pp 133ndash149 2008

[12] J H Zheng Value Distribution of Meromorphic FunctionsTsinghua University Press Beijing China 2010

[13] C H Li and Y X Gu ldquoA fundamental inequality for 119870-quasimeromorphic mappings in an angular domain and itsapplicationrdquo Acta Mathematica Sinica vol 49 no 6 pp 1279ndash1286 2006 (Chinese)

[14] Z-X Xuan ldquoOn the existence of 119879-direction of algebroidfunctions a problem of J H Zhengrdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 540ndash547 2008

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Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

Proof Put 119863119903 = 119911 isin C 1 lt |119911| lt 119903 cap Ω(120579 120593) and 1198650 = 119870 minus119886V Using Lemma 4 we have

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2) [119878 (119903 Ω (120579 120593) 119891)

minus119878 (1 Ω (120579 120593) 119891)]

le

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V) + 119867119871 (119903)

(23)

where119867 = (119862 + 9119902ℎ)1198971205753 which depends only on 1198650 that isonly on 119886V V = 1 2 119902 and

119871 (119903) = 119871 (119903 120579 minus 120593 120579 + 120593) + 119871 (1 120579 minus 120593 120579 + 120593)

+ 119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)

le 119871 (119903 120579 minus 120575 120579 + 120575) + 119871 (1 120579 minus 120575 120579 + 120575)

+ 119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)

(24)

Hence

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)

times [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

minus

119902

sumV=1

119897V119897V + 1

119899119897V) (119903 120579 120575 119886V) minus 119867119871 (119903 120579 minus 120575 120579 + 120575)

minus 119867119871 (1 120579 minus 120575 120579 + 120575)

le 119867 [119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]

(25)

Denote the left expression of (25) by 119860(119903 120593) thus

119889 (119860 (119903 120593))

119889120593

= (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)

times119889 [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

119889120593

(26)

We claim the fact that

[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]2le

2120587

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

times119889 (119860 (119903 120593))

119889120593log 119903

(27)

In fact it follows from the definition of 119871(119903 120595) and Schwarzrsquosinequality that

[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]2

le 2[

[

(int119903

1

100381610038161003816100381610038161198911015840(119905119890119894(120579minus120593))

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894(120579minus120593))

10038161003816100381610038162)119889119905)

2

+ (int119903

1

100381610038161003816100381610038161198911015840(119905119890119894(120579+120593))

10038161003816100381610038161003816

(1 +1003816100381610038161003816119891(119905119890119894(120579+120593))

10038161003816100381610038162)119889119905)

2

]

]

le 2120587119889 [119878 (119903 Ω (120579 120593) 119891) minus 119878 (1 Ω (120579 120593) 119891)]

119889120593log 119903

=2120587

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

119889 (119860 (119903 120593))

119889120593log 119903

(28)

Noting119860(119903 120593) is an increasing function of 120593 we see that thenthere exists a 1205750 gt 0 such that 119860(119903 120593) gt 0 when 120593 gt 1205750 and119860(119903 120593) le 0 when 120593 le 1205750 For 120593 gt 1205750 by (25) and (27)

[119860 (119903 120593)]2le 1198672[119871 (119903 120579 minus 120593) + 119871 (119903 120579 + 120593)]

2

le21205871198672

(sum119902

119895=1(1 minus

1

119897V + 1) minus 2)

log 119903119889 (119860 (119903 120593))

119889120593

(29)

that is

119889120593 le21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2)

log 119903119889 (119860 (119903 120593))

[119860 (119903 120593)]2 (30)

Integrating each side of the inequality leads to

120575 minus 120593 = int120575

120593

119889120593 le21205871198672

(sum119902

119895=1(1 minus

1

119897V + 1) minus 2)119860 (119903 120593)

log 119903

(31)

Thus

119860 (119903 120593) le21205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120591)

log 119903 (32)

On the case of 120593 le 1205750 the above inequality is obviously validbecause of 119860(119903 120593) le 0 Replacing 119860(119903 120593) in the above

The Scientific World Journal 5

inequality with its explicit expression we see that (21) is esta-blished Therefore

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+1205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log2119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (1Ω (120579 120593) 119891)

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891) log 119903

+ 119867119871 (1 120579 minus 120575 120579 + 120575) log 119903 + 120594 (119903 120579 minus 120575 120579 + 120575) (33)

where 120594(119903 120579 minus 120575 120579 + 120575) = 119867int119903

1(119871(119905 120579 minus 120575 120579 + 120575)119905)119889119905

Lemma 6 (Zhang [7]) Under the condition ofTheorem 5 onehas

120594 (119903 120579 minus 120575 120579 + 120575) = 119867int119903

1

119871 (119905 120579 minus 120575 120579 + 120575)

119905119889119905

le 119867radic2120575120587119878 (119903 Ω (120579 120575) 119891) log 119903(34)

or

120594 (119903 120579 minus 120575 120579 + 120575) le 119867radic2120575120587119879 (119903 Ω (120579 120575) 119891)

times log119879 (119903 Ω (120579 120575) 119891)

(35)

with at most one exceptional set 119864120575 of 119903 where 119864120575 consists of aseries of intervals and satisfies

int119864120575

1

119903 log 119903119889119903 le

1

log119879 (119903 Ω (120579 120575) 119891)lt infin (36)

In particular if the order of 119891(119911) is 120588 (0 lt 120588 lt +infin) then

120594 (119903 120579 minus 120575 120579 + 120575) le 119874 (11990331205884) (37)

FromTheorem 3 and Lemma 6 we can write the result inTheorem 3 as

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+ 119874 (log2119903) + 120594 (119903 120579 minus 120575 120579 + 120575)

(38)

If the order of119891(119911) is120588 (0 lt 120588 lt +infin) then the inequalitywillbe

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V) + 119874 (11990331205884)

(39)

4 Bloch Direction of Meromorphic Functions

In this section we will research the singular direction corre-sponding toTheoremA Suppose that119891(119911) is a meromorphicfunction of infinite order Then there is a real function 120588(119903)called anHiongrsquos proximate order (see [8]) of119891(119911) which hasthe following properties (i) 120588(119903) is continuous and nonde-creasing for 119903 ge 1199030 (1199030 gt 0) and tends to +infin as 119903 rarr +infin (ii)The function 119880(119903) = 119903120588(119903)(119903 ge 1199030) satisfies the condition

lim119903rarr+infin

log119880 (119877)

log119880 (119903)= 1 119877 = 119903 +

119903

log119880 (119903)

lim sup119903rarr+infin

log119879 (119903 119891)log119880 (119903)

= 1

(40)

For a meromorphic function of infinite order ZhuangQitai (or Chuang Chitai) [9] gives the following definition ofBorel direction and Bloch direction

Definition 7 Let 119891(119911) be a meromorphic function of infiniteorder and 120588(119903) an order of119891(119911) A direction arg 119911 = 120579 is calleda Borel direction of order 120588(119903) of 119891(119911) if no matter how smallthe positive number 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)120588 (119903) log 119903

= 1 (41)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588(119903) of119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that119896119895 is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (42)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)120588 (119903) log 119903

= 1 (43)

For the connection of Borel direction and Bloch directionof meromorphic function of infinite order Chuang [9] hasproved the following theorem

Theorem C Let 119891(119911) be a meromorphic function of infiniteorder and 120588(119903) an order of 119891(119911) Then every Borel direction oforder 120588(119903) of 119891(119911) is a Bloch direction of order 120588(119903) of 119891(119911)

6 The Scientific World Journal

It is natural to consider whether there exists a similarresult if meromorphic function of order infinity is replacedwithmeromorphic function of order 120588 (0 lt 120588 lt +infin) In thissection we extend the above theorem to meromorphic func-tion of order 120588 (0 lt 120588 lt +infin)

Definition 8 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) A direction arg 119911 = 120579 is called a Borel dir-ection of order 120588 of 119891(119911) if no matter how small the positivenumber 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)log 119903

= 120588 (44)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588 of 119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (45)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)log 119903

= 120588 (46)

Theorem 9 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then every Borel direction of order 120588 of 119891(119911)is a Bloch direction of order 120588 of 119891(119911)

In order to prove Theorem 9 we need the followinglemma

Lemma 10 (Zhang [7]) Let 119891(119911) be a meromorphic functionof order 120588 (0 lt 120588 lt +infin) Then a direction arg 119911 = 120579 is a Boreldirection of order 120588 of 119891(119911) if and only if it satisfies

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (47)

for any 120576 (0 lt 120576 lt 1205872)

We are now in the position to proveTheorem 9

Proof Suppose that arg 119911 = 120579 is a Borel direction of order 120588 of119891(119911) then for any 120576 (0 lt 120576 lt 1205872) we have

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (48)

If arg 119911 = 120579 is not a Bloch direction of order 120588 of 119891(119911) thenthere exit a system 119886119895 (119895 = 1 2 119902) of distinct values and asystem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (49)

And for any integer 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (50)

Hence we can get

lim sup119903rarrinfin

log119873119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (51)

for any integer 119895 (1 le 119895 le 119902)Therefore we can find a positivenumber 120591 lt 120588 such that

119873119896119895) (119903 120579 2120576 119886119895) le 119903120591 (52)

By (39) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Ω (120579 120576) 119891)

le

119902

sumV=1

119896119895

119896119895 + 1119873119896119895) (119903 120579 2120576 119886V) + 119874 (11990331205884)

le 119874 (119903120577)

(53)

where 120577 = max120591 31205884 lt 120588Hence

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120577 lt 120588 (54)

This contradicts with (48) andTheorem 9 follows

Corollary 11 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then there is a direction arg 119911 = 120579 which is aBloch direction of order 120588 of 119891(119911)

Note that Corollary 11 is a corresponding result of Theo-rem A in angular distribution

5 Pseudo-T Direction ofMeromorphic Functions

In 2003 Zheng [1] introduced a new singular direction calledT direction We call 119869 arg 119911 = 120579 the T direction of 119891(119911)provided that given any 119886 isin Cinfin possiblywith exception of atmost two values of 119886 for any positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873(119903 120579 120576 119886)

119879 (119903 119891)gt 0 (55)

For the existence of T direction of meromorphic function119891(119911) Guo et al [10] proved the followingTheorem

Theorem C Let 119891(119911) be a meromorphic function and satisfy

lim sup119903rarrinfin

119879 (119903 119891)

log2119903= infin (56)

Then 119891(119911)must have a T direction

The Scientific World Journal 7

Theorem C was conjectured by Zheng [1] In [11] theauthors study the existence of T direction of 119891(119911) concerningmultiple values We call 119869 arg 119911 = 120579 the T direction of 119891(119911)concerningmultiple values provided that given any 119886 isin Cinfinpossibly with exception of at most [(2119897 + 2)119897] values of 119886 forany positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873119897) (119903 120579 120576 119886)

119879 (119903 119891)gt 0 (57)

where [119909] implies the maximum integer number which doesnot exceed 119909 and 119897 is a positive integer

Theorem D Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a T direction of 119891(119911) concerningmultiple values

Note that the T direction of meromorphic function con-cerning multiple values is a refinement of the ordinary T dir-ection since [(2119897+2)119897] rarr 2 as 119897 rarr infin Since Zheng [1] gavethe definition of T direction then there is a considerablenumber result related this directionwe refer the reader to [12]for finding a careful discussion of this direction

It is well known that T direction is a concept in angulardistribution which corresponds to the Nevanlinna secondfundamental theorem in module distribution It is natural toconsider the corresponding result to Theorem B in angulardistribution

Definition 12 Let 119891(119911) be a meromorphic function A direc-tion arg 119911 = 120579 is called a pseudo-T direction of119891(119911) if for anynumber 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (58)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 120579 120576 119886119895)

119879 (119903 119891)gt 0 (59)

Theorem 13 Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a pseudo-T direction of 119891(119911)

Remark 14 (i) In Theorem C 119902 = 3 119896119895 = infin (119895 = 1 2 3) soTheorem C is a special case of Theorem 13

(ii) If 119896119895 = 1 (119895 = 1 2 119902) then 119902 = 5 if 119896119895 = 2 (119895 =1 2 119902) then 119902 = 4 if 119896119895 = 119897 ge 3 (119895 = 1 2 119902) then119902 = 3 So Theorem D is a special case of Theorem 13

In order to prove Theorem 13 we need the followinglemma

Lemma 15 (Li and Gu [13] see also Xuan [14]) Suppose thatΨ(119903) is a nonnegative increasing function in (1infin) and satisfies

lim sup119903rarrinfin

Ψ (119903)

log2119903= infin (60)

Then for any set 119864 sub (1infin) such that int119864(1119903 log 119903)119889119903 lt 13

one has

lim sup119903rarrinfin119903isin(1infin)minus119864

Ψ (119903)

log2119903= infin (61)

Proof Firstly we prove the following statement Let 119898 (119898 ge4) be a fixed positive integer 1205790 = 0 1205791 = 2120587119898 120579119898minus1 =(119898 minus 1)2120587119898 120579119898 = 1205790 We put Δ(120579119894) = 119911 | arg 119911 minus 120579119894| lt2120587119898 Δ119900(120579119894) = 119911 | arg 119911 minus 120579119894| lt 120587119898 119894 = 0 1 119898 minus 1Δ(120579119898) = Δ(1205790) Δ

119900(120579119898) = Δ119900(1205790) Then among these119898 angu-lar domains Δ(120579119894) there is at least an angular domain Δ(120579119894)such that for any system 119886119895 (119895 = 1 2 119902) of distinct valuesand any system 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positiveinteger or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (62)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (63)

Otherwise for any angular domainΔ(120579119894) (1 le 119894 le 119898) there isa system 119886

119895

119894(119895 = 1 2 119902) of distinct values and a system

119896119895

119894(119895 = 1 2 119902) such that 119896119895

119894is a positive integer or+infin and

that

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

) gt 2 (64)

for any 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

119873119896119895

119894) (119903 Δ (120579119894) 119886

119895

119894)

119879 (119903 119891)= 0 (65)

Put

119902

sum119895=1

(1 minus1

119896119895 + 1) = min1le119894le119898

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

)

gt 2 (66)

ApplyingTheorem 5 to Δ119900(120579119894+1) Δ(120579119894+1) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Δ119900 (120579119894+1) 119891)

le

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895

119894) (119903 Δ (120579119894+1) 119886

119895

119894+1)

+ 119874 (log2119903) + 120594 (119903 Δ (120579119894+1))

(67)

8 The Scientific World Journal

Noting119879(119903 119891) = sum119898minus1

119894=0119879(119903 Δ119900(120579119894+1) 119891) and adding two sides

of the above expression from 119894 = 0 to119898 minus 1 we can obtain

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 119891)

le119898minus1

sum119894=0

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895) (119903 Δ (120579119894+1) 119886119895

119894+1)

+ 119874 (log2119903) +119898minus1

sum119894=0

120594 (119903 Δ (120579119894+1))

(68)

For any 119894 there exists a 119903119894 the inequality 119879(119903 Δ119900(120579119894+1) 119891) gt

1198903119898 would bold for 119903 gt 119903119894 while the inequality (22) does notlook appropriate here Put 119864Δ119900(120579119894+1) is the set of 119903 which con-sists of a series of intervals and satisfies

int119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt

1

3119898 (69)

Let 1199030 = max119903119894 119894 = 1 2 119898 we have for any 119894 119879(1199030Δ119900(120579119894+1) 119891) gt 1198903119898 then

intcup119898minus1119894=0119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le119898minus1

sum119894=0

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt1

3

(70)

Applying Lemma 15 we have

lim sup119903rarrinfin119903isin(1infin)minus119864

119879 (119903 119891)

log2119903= infin (71)

where 119864 = cup119898minus1119894=0

119864Δ119900(120579119894+1) Therefore there exists a sequence1199031015840119899isin (1infin) minus 119864 such that

lim119899rarrinfin

119879 (1199031015840119899 119891)

log21199031015840119899

= infin (72)

It follows from (38) (68) and (72) that

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2) le 0 (73)

Hence119902

sum119895=1

(1 minus1

119896119895 + 1) le 2 (74)

This is a contradiction Hence for an arbitrary positive inte-ger 119898 there is at least an angular domain Δ(120579119894) such that forany system 119886119895 (119895 = 1 2 119902) of distinct values and any sys-tem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (75)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (76)

Choosing subsequence of 120579119898 still denote it 120579119898 we assumethat 120579119898 rarr 1205790 Put 119871 arg 119911 = 1205790 then 119871 is a pseudo-Tdirection that is stated in Definition 12

In fact for any 120576 (0 lt 120576 lt 1205872) when 119898 is sufficientlylarge we have Δ(120579119898) sub Ω(1205790 120576) By (76) we have

lim sup119903rarrinfin

119873119896119895) (119903 1205790 120576 119886119895)

119879 (119903 119891)ge lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119898) 119886119895)

119879 (119903 119891)gt 0

(77)

Hence Theorem 13 holds in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The first author was partly supported by the National NaturalScience Foundation of China (Grant no 11201395) and by theScience Foundation of Educational Commission of HubeiProvince (Grant no Q20132801) The second author waspartly supported by the NSF of Jiangxi Province (Grant20122BAB201006) The third author was partly supported byBeijing Natural Science Foundation (Grant no 1132013)

References

[1] J H Zheng ldquoOn transcendental meromorphic functions withradially distributed valuesrdquo Science in China A vol 47 no 3 pp401ndash416 2004

[2] L Yang Value Distribution Theory Springer Berlin Germany1993

[3] M Tsuji PotentialTheory inModern FunctionTheoryMaruzenTokyo Japan 1959

[4] D C Sun ldquoMain theorem on covering surfacesrdquo Acta Mathe-matica Scientia B vol 14 no 2 pp 213ndash225 1994

[5] D C Sun ldquoInequalities for covering surfaces and their applica-tionsrdquoChinese Annals ofMathematics A vol 18 no 1 pp 59ndash641997 (Chinese)

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs Clarendon Press Oxford UK 1964

[7] X L Zhang ldquoA fundamental inequality for meromorphic func-tions in an angular domain and its applicationrdquoActaMathemat-ica Sinica vol 10 no 3 pp 308ndash314 1994

[8] Q T Zhuang Singular Directions of Meromorphic Functionsvol 11 of Series of Monographs in Pure and AppliedMathematicsScience Press Beijing China 1982 (Chinese)

[9] C T Chuang ldquoOn Borel directions of meromorphic functionsof infinite order IIrdquoBulletin of theHongKongMathematical Soc-iety vol 2 no 2 pp 305ndash323 1999

[10] H Guo J H Zheng and TW Ng ldquoOn a new singular directionof meromorphic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 69 no 2 pp 277ndash287 2004

The Scientific World Journal 9

[11] Z-J Wu and D-C Sun ldquoOn the existence of 119879 direction ofmeromorphic function concerning multiple valuesrdquo KodaiMathematical Journal vol 31 no 1 pp 133ndash149 2008

[12] J H Zheng Value Distribution of Meromorphic FunctionsTsinghua University Press Beijing China 2010

[13] C H Li and Y X Gu ldquoA fundamental inequality for 119870-quasimeromorphic mappings in an angular domain and itsapplicationrdquo Acta Mathematica Sinica vol 49 no 6 pp 1279ndash1286 2006 (Chinese)

[14] Z-X Xuan ldquoOn the existence of 119879-direction of algebroidfunctions a problem of J H Zhengrdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 540ndash547 2008

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Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

inequality with its explicit expression we see that (21) is esta-blished Therefore

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+1205871198672

(sum119902

119895=1(1 minus 1 (119897V + 1)) minus 2) (120575 minus 120593)

log2119903

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (1Ω (120579 120593) 119891)

+ (

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119878 (1 Ω (120579 120593) 119891) log 119903

+ 119867119871 (1 120579 minus 120575 120579 + 120575) log 119903 + 120594 (119903 120579 minus 120575 120579 + 120575) (33)

where 120594(119903 120579 minus 120575 120579 + 120575) = 119867int119903

1(119871(119905 120579 minus 120575 120579 + 120575)119905)119889119905

Lemma 6 (Zhang [7]) Under the condition ofTheorem 5 onehas

120594 (119903 120579 minus 120575 120579 + 120575) = 119867int119903

1

119871 (119905 120579 minus 120575 120579 + 120575)

119905119889119905

le 119867radic2120575120587119878 (119903 Ω (120579 120575) 119891) log 119903(34)

or

120594 (119903 120579 minus 120575 120579 + 120575) le 119867radic2120575120587119879 (119903 Ω (120579 120575) 119891)

times log119879 (119903 Ω (120579 120575) 119891)

(35)

with at most one exceptional set 119864120575 of 119903 where 119864120575 consists of aseries of intervals and satisfies

int119864120575

1

119903 log 119903119889119903 le

1

log119879 (119903 Ω (120579 120575) 119891)lt infin (36)

In particular if the order of 119891(119911) is 120588 (0 lt 120588 lt +infin) then

120594 (119903 120579 minus 120575 120579 + 120575) le 119874 (11990331205884) (37)

FromTheorem 3 and Lemma 6 we can write the result inTheorem 3 as

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V)

+ 119874 (log2119903) + 120594 (119903 120579 minus 120575 120579 + 120575)

(38)

If the order of119891(119911) is120588 (0 lt 120588 lt +infin) then the inequalitywillbe

(

119902

sum119895=1

(1 minus1

119897V + 1) minus 2)119879 (119903 Ω (120579 120593) 119891)

le

119902

sumV=1

119897V119897V + 1

119873119897V) (119903 120579 120575 119886V) + 119874 (11990331205884)

(39)

4 Bloch Direction of Meromorphic Functions

In this section we will research the singular direction corre-sponding toTheoremA Suppose that119891(119911) is a meromorphicfunction of infinite order Then there is a real function 120588(119903)called anHiongrsquos proximate order (see [8]) of119891(119911) which hasthe following properties (i) 120588(119903) is continuous and nonde-creasing for 119903 ge 1199030 (1199030 gt 0) and tends to +infin as 119903 rarr +infin (ii)The function 119880(119903) = 119903120588(119903)(119903 ge 1199030) satisfies the condition

lim119903rarr+infin

log119880 (119877)

log119880 (119903)= 1 119877 = 119903 +

119903

log119880 (119903)

lim sup119903rarr+infin

log119879 (119903 119891)log119880 (119903)

= 1

(40)

For a meromorphic function of infinite order ZhuangQitai (or Chuang Chitai) [9] gives the following definition ofBorel direction and Bloch direction

Definition 7 Let 119891(119911) be a meromorphic function of infiniteorder and 120588(119903) an order of119891(119911) A direction arg 119911 = 120579 is calleda Borel direction of order 120588(119903) of 119891(119911) if no matter how smallthe positive number 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)120588 (119903) log 119903

= 1 (41)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588(119903) of119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that119896119895 is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (42)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)120588 (119903) log 119903

= 1 (43)

For the connection of Borel direction and Bloch directionof meromorphic function of infinite order Chuang [9] hasproved the following theorem

Theorem C Let 119891(119911) be a meromorphic function of infiniteorder and 120588(119903) an order of 119891(119911) Then every Borel direction oforder 120588(119903) of 119891(119911) is a Bloch direction of order 120588(119903) of 119891(119911)

6 The Scientific World Journal

It is natural to consider whether there exists a similarresult if meromorphic function of order infinity is replacedwithmeromorphic function of order 120588 (0 lt 120588 lt +infin) In thissection we extend the above theorem to meromorphic func-tion of order 120588 (0 lt 120588 lt +infin)

Definition 8 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) A direction arg 119911 = 120579 is called a Borel dir-ection of order 120588 of 119891(119911) if no matter how small the positivenumber 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)log 119903

= 120588 (44)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588 of 119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (45)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)log 119903

= 120588 (46)

Theorem 9 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then every Borel direction of order 120588 of 119891(119911)is a Bloch direction of order 120588 of 119891(119911)

In order to prove Theorem 9 we need the followinglemma

Lemma 10 (Zhang [7]) Let 119891(119911) be a meromorphic functionof order 120588 (0 lt 120588 lt +infin) Then a direction arg 119911 = 120579 is a Boreldirection of order 120588 of 119891(119911) if and only if it satisfies

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (47)

for any 120576 (0 lt 120576 lt 1205872)

We are now in the position to proveTheorem 9

Proof Suppose that arg 119911 = 120579 is a Borel direction of order 120588 of119891(119911) then for any 120576 (0 lt 120576 lt 1205872) we have

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (48)

If arg 119911 = 120579 is not a Bloch direction of order 120588 of 119891(119911) thenthere exit a system 119886119895 (119895 = 1 2 119902) of distinct values and asystem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (49)

And for any integer 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (50)

Hence we can get

lim sup119903rarrinfin

log119873119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (51)

for any integer 119895 (1 le 119895 le 119902)Therefore we can find a positivenumber 120591 lt 120588 such that

119873119896119895) (119903 120579 2120576 119886119895) le 119903120591 (52)

By (39) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Ω (120579 120576) 119891)

le

119902

sumV=1

119896119895

119896119895 + 1119873119896119895) (119903 120579 2120576 119886V) + 119874 (11990331205884)

le 119874 (119903120577)

(53)

where 120577 = max120591 31205884 lt 120588Hence

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120577 lt 120588 (54)

This contradicts with (48) andTheorem 9 follows

Corollary 11 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then there is a direction arg 119911 = 120579 which is aBloch direction of order 120588 of 119891(119911)

Note that Corollary 11 is a corresponding result of Theo-rem A in angular distribution

5 Pseudo-T Direction ofMeromorphic Functions

In 2003 Zheng [1] introduced a new singular direction calledT direction We call 119869 arg 119911 = 120579 the T direction of 119891(119911)provided that given any 119886 isin Cinfin possiblywith exception of atmost two values of 119886 for any positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873(119903 120579 120576 119886)

119879 (119903 119891)gt 0 (55)

For the existence of T direction of meromorphic function119891(119911) Guo et al [10] proved the followingTheorem

Theorem C Let 119891(119911) be a meromorphic function and satisfy

lim sup119903rarrinfin

119879 (119903 119891)

log2119903= infin (56)

Then 119891(119911)must have a T direction

The Scientific World Journal 7

Theorem C was conjectured by Zheng [1] In [11] theauthors study the existence of T direction of 119891(119911) concerningmultiple values We call 119869 arg 119911 = 120579 the T direction of 119891(119911)concerningmultiple values provided that given any 119886 isin Cinfinpossibly with exception of at most [(2119897 + 2)119897] values of 119886 forany positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873119897) (119903 120579 120576 119886)

119879 (119903 119891)gt 0 (57)

where [119909] implies the maximum integer number which doesnot exceed 119909 and 119897 is a positive integer

Theorem D Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a T direction of 119891(119911) concerningmultiple values

Note that the T direction of meromorphic function con-cerning multiple values is a refinement of the ordinary T dir-ection since [(2119897+2)119897] rarr 2 as 119897 rarr infin Since Zheng [1] gavethe definition of T direction then there is a considerablenumber result related this directionwe refer the reader to [12]for finding a careful discussion of this direction

It is well known that T direction is a concept in angulardistribution which corresponds to the Nevanlinna secondfundamental theorem in module distribution It is natural toconsider the corresponding result to Theorem B in angulardistribution

Definition 12 Let 119891(119911) be a meromorphic function A direc-tion arg 119911 = 120579 is called a pseudo-T direction of119891(119911) if for anynumber 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (58)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 120579 120576 119886119895)

119879 (119903 119891)gt 0 (59)

Theorem 13 Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a pseudo-T direction of 119891(119911)

Remark 14 (i) In Theorem C 119902 = 3 119896119895 = infin (119895 = 1 2 3) soTheorem C is a special case of Theorem 13

(ii) If 119896119895 = 1 (119895 = 1 2 119902) then 119902 = 5 if 119896119895 = 2 (119895 =1 2 119902) then 119902 = 4 if 119896119895 = 119897 ge 3 (119895 = 1 2 119902) then119902 = 3 So Theorem D is a special case of Theorem 13

In order to prove Theorem 13 we need the followinglemma

Lemma 15 (Li and Gu [13] see also Xuan [14]) Suppose thatΨ(119903) is a nonnegative increasing function in (1infin) and satisfies

lim sup119903rarrinfin

Ψ (119903)

log2119903= infin (60)

Then for any set 119864 sub (1infin) such that int119864(1119903 log 119903)119889119903 lt 13

one has

lim sup119903rarrinfin119903isin(1infin)minus119864

Ψ (119903)

log2119903= infin (61)

Proof Firstly we prove the following statement Let 119898 (119898 ge4) be a fixed positive integer 1205790 = 0 1205791 = 2120587119898 120579119898minus1 =(119898 minus 1)2120587119898 120579119898 = 1205790 We put Δ(120579119894) = 119911 | arg 119911 minus 120579119894| lt2120587119898 Δ119900(120579119894) = 119911 | arg 119911 minus 120579119894| lt 120587119898 119894 = 0 1 119898 minus 1Δ(120579119898) = Δ(1205790) Δ

119900(120579119898) = Δ119900(1205790) Then among these119898 angu-lar domains Δ(120579119894) there is at least an angular domain Δ(120579119894)such that for any system 119886119895 (119895 = 1 2 119902) of distinct valuesand any system 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positiveinteger or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (62)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (63)

Otherwise for any angular domainΔ(120579119894) (1 le 119894 le 119898) there isa system 119886

119895

119894(119895 = 1 2 119902) of distinct values and a system

119896119895

119894(119895 = 1 2 119902) such that 119896119895

119894is a positive integer or+infin and

that

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

) gt 2 (64)

for any 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

119873119896119895

119894) (119903 Δ (120579119894) 119886

119895

119894)

119879 (119903 119891)= 0 (65)

Put

119902

sum119895=1

(1 minus1

119896119895 + 1) = min1le119894le119898

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

)

gt 2 (66)

ApplyingTheorem 5 to Δ119900(120579119894+1) Δ(120579119894+1) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Δ119900 (120579119894+1) 119891)

le

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895

119894) (119903 Δ (120579119894+1) 119886

119895

119894+1)

+ 119874 (log2119903) + 120594 (119903 Δ (120579119894+1))

(67)

8 The Scientific World Journal

Noting119879(119903 119891) = sum119898minus1

119894=0119879(119903 Δ119900(120579119894+1) 119891) and adding two sides

of the above expression from 119894 = 0 to119898 minus 1 we can obtain

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 119891)

le119898minus1

sum119894=0

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895) (119903 Δ (120579119894+1) 119886119895

119894+1)

+ 119874 (log2119903) +119898minus1

sum119894=0

120594 (119903 Δ (120579119894+1))

(68)

For any 119894 there exists a 119903119894 the inequality 119879(119903 Δ119900(120579119894+1) 119891) gt

1198903119898 would bold for 119903 gt 119903119894 while the inequality (22) does notlook appropriate here Put 119864Δ119900(120579119894+1) is the set of 119903 which con-sists of a series of intervals and satisfies

int119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt

1

3119898 (69)

Let 1199030 = max119903119894 119894 = 1 2 119898 we have for any 119894 119879(1199030Δ119900(120579119894+1) 119891) gt 1198903119898 then

intcup119898minus1119894=0119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le119898minus1

sum119894=0

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt1

3

(70)

Applying Lemma 15 we have

lim sup119903rarrinfin119903isin(1infin)minus119864

119879 (119903 119891)

log2119903= infin (71)

where 119864 = cup119898minus1119894=0

119864Δ119900(120579119894+1) Therefore there exists a sequence1199031015840119899isin (1infin) minus 119864 such that

lim119899rarrinfin

119879 (1199031015840119899 119891)

log21199031015840119899

= infin (72)

It follows from (38) (68) and (72) that

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2) le 0 (73)

Hence119902

sum119895=1

(1 minus1

119896119895 + 1) le 2 (74)

This is a contradiction Hence for an arbitrary positive inte-ger 119898 there is at least an angular domain Δ(120579119894) such that forany system 119886119895 (119895 = 1 2 119902) of distinct values and any sys-tem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (75)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (76)

Choosing subsequence of 120579119898 still denote it 120579119898 we assumethat 120579119898 rarr 1205790 Put 119871 arg 119911 = 1205790 then 119871 is a pseudo-Tdirection that is stated in Definition 12

In fact for any 120576 (0 lt 120576 lt 1205872) when 119898 is sufficientlylarge we have Δ(120579119898) sub Ω(1205790 120576) By (76) we have

lim sup119903rarrinfin

119873119896119895) (119903 1205790 120576 119886119895)

119879 (119903 119891)ge lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119898) 119886119895)

119879 (119903 119891)gt 0

(77)

Hence Theorem 13 holds in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The first author was partly supported by the National NaturalScience Foundation of China (Grant no 11201395) and by theScience Foundation of Educational Commission of HubeiProvince (Grant no Q20132801) The second author waspartly supported by the NSF of Jiangxi Province (Grant20122BAB201006) The third author was partly supported byBeijing Natural Science Foundation (Grant no 1132013)

References

[1] J H Zheng ldquoOn transcendental meromorphic functions withradially distributed valuesrdquo Science in China A vol 47 no 3 pp401ndash416 2004

[2] L Yang Value Distribution Theory Springer Berlin Germany1993

[3] M Tsuji PotentialTheory inModern FunctionTheoryMaruzenTokyo Japan 1959

[4] D C Sun ldquoMain theorem on covering surfacesrdquo Acta Mathe-matica Scientia B vol 14 no 2 pp 213ndash225 1994

[5] D C Sun ldquoInequalities for covering surfaces and their applica-tionsrdquoChinese Annals ofMathematics A vol 18 no 1 pp 59ndash641997 (Chinese)

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs Clarendon Press Oxford UK 1964

[7] X L Zhang ldquoA fundamental inequality for meromorphic func-tions in an angular domain and its applicationrdquoActaMathemat-ica Sinica vol 10 no 3 pp 308ndash314 1994

[8] Q T Zhuang Singular Directions of Meromorphic Functionsvol 11 of Series of Monographs in Pure and AppliedMathematicsScience Press Beijing China 1982 (Chinese)

[9] C T Chuang ldquoOn Borel directions of meromorphic functionsof infinite order IIrdquoBulletin of theHongKongMathematical Soc-iety vol 2 no 2 pp 305ndash323 1999

[10] H Guo J H Zheng and TW Ng ldquoOn a new singular directionof meromorphic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 69 no 2 pp 277ndash287 2004

The Scientific World Journal 9

[11] Z-J Wu and D-C Sun ldquoOn the existence of 119879 direction ofmeromorphic function concerning multiple valuesrdquo KodaiMathematical Journal vol 31 no 1 pp 133ndash149 2008

[12] J H Zheng Value Distribution of Meromorphic FunctionsTsinghua University Press Beijing China 2010

[13] C H Li and Y X Gu ldquoA fundamental inequality for 119870-quasimeromorphic mappings in an angular domain and itsapplicationrdquo Acta Mathematica Sinica vol 49 no 6 pp 1279ndash1286 2006 (Chinese)

[14] Z-X Xuan ldquoOn the existence of 119879-direction of algebroidfunctions a problem of J H Zhengrdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 540ndash547 2008

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

It is natural to consider whether there exists a similarresult if meromorphic function of order infinity is replacedwithmeromorphic function of order 120588 (0 lt 120588 lt +infin) In thissection we extend the above theorem to meromorphic func-tion of order 120588 (0 lt 120588 lt +infin)

Definition 8 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) A direction arg 119911 = 120579 is called a Borel dir-ection of order 120588 of 119891(119911) if no matter how small the positivenumber 120578 is for each value 120596 one has

lim sup119903rarrinfin

log 119899 (119903 120579 120578 120596)log 119903

= 120588 (44)

except for at most two exceptional values 120596 A directionarg 119911 = 120579 is called a Bloch direction of order 120588 of 119891(119911) if forany number 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (45)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 120576 119886119895)log 119903

= 120588 (46)

Theorem 9 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then every Borel direction of order 120588 of 119891(119911)is a Bloch direction of order 120588 of 119891(119911)

In order to prove Theorem 9 we need the followinglemma

Lemma 10 (Zhang [7]) Let 119891(119911) be a meromorphic functionof order 120588 (0 lt 120588 lt +infin) Then a direction arg 119911 = 120579 is a Boreldirection of order 120588 of 119891(119911) if and only if it satisfies

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (47)

for any 120576 (0 lt 120576 lt 1205872)

We are now in the position to proveTheorem 9

Proof Suppose that arg 119911 = 120579 is a Borel direction of order 120588 of119891(119911) then for any 120576 (0 lt 120576 lt 1205872) we have

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120588 (48)

If arg 119911 = 120579 is not a Bloch direction of order 120588 of 119891(119911) thenthere exit a system 119886119895 (119895 = 1 2 119902) of distinct values and asystem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (49)

And for any integer 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

log 119899119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (50)

Hence we can get

lim sup119903rarrinfin

log119873119896119895) (119903 120579 2120576 119886119895)log 119903

lt 120588 (51)

for any integer 119895 (1 le 119895 le 119902)Therefore we can find a positivenumber 120591 lt 120588 such that

119873119896119895) (119903 120579 2120576 119886119895) le 119903120591 (52)

By (39) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Ω (120579 120576) 119891)

le

119902

sumV=1

119896119895

119896119895 + 1119873119896119895) (119903 120579 2120576 119886V) + 119874 (11990331205884)

le 119874 (119903120577)

(53)

where 120577 = max120591 31205884 lt 120588Hence

lim sup119903rarrinfin

log119879 (119903 Ω (120579 120576) 119891)

log 119903= 120577 lt 120588 (54)

This contradicts with (48) andTheorem 9 follows

Corollary 11 Let 119891(119911) be a meromorphic function of order120588 (0 lt 120588 lt +infin) Then there is a direction arg 119911 = 120579 which is aBloch direction of order 120588 of 119891(119911)

Note that Corollary 11 is a corresponding result of Theo-rem A in angular distribution

5 Pseudo-T Direction ofMeromorphic Functions

In 2003 Zheng [1] introduced a new singular direction calledT direction We call 119869 arg 119911 = 120579 the T direction of 119891(119911)provided that given any 119886 isin Cinfin possiblywith exception of atmost two values of 119886 for any positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873(119903 120579 120576 119886)

119879 (119903 119891)gt 0 (55)

For the existence of T direction of meromorphic function119891(119911) Guo et al [10] proved the followingTheorem

Theorem C Let 119891(119911) be a meromorphic function and satisfy

lim sup119903rarrinfin

119879 (119903 119891)

log2119903= infin (56)

Then 119891(119911)must have a T direction

The Scientific World Journal 7

Theorem C was conjectured by Zheng [1] In [11] theauthors study the existence of T direction of 119891(119911) concerningmultiple values We call 119869 arg 119911 = 120579 the T direction of 119891(119911)concerningmultiple values provided that given any 119886 isin Cinfinpossibly with exception of at most [(2119897 + 2)119897] values of 119886 forany positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873119897) (119903 120579 120576 119886)

119879 (119903 119891)gt 0 (57)

where [119909] implies the maximum integer number which doesnot exceed 119909 and 119897 is a positive integer

Theorem D Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a T direction of 119891(119911) concerningmultiple values

Note that the T direction of meromorphic function con-cerning multiple values is a refinement of the ordinary T dir-ection since [(2119897+2)119897] rarr 2 as 119897 rarr infin Since Zheng [1] gavethe definition of T direction then there is a considerablenumber result related this directionwe refer the reader to [12]for finding a careful discussion of this direction

It is well known that T direction is a concept in angulardistribution which corresponds to the Nevanlinna secondfundamental theorem in module distribution It is natural toconsider the corresponding result to Theorem B in angulardistribution

Definition 12 Let 119891(119911) be a meromorphic function A direc-tion arg 119911 = 120579 is called a pseudo-T direction of119891(119911) if for anynumber 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (58)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 120579 120576 119886119895)

119879 (119903 119891)gt 0 (59)

Theorem 13 Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a pseudo-T direction of 119891(119911)

Remark 14 (i) In Theorem C 119902 = 3 119896119895 = infin (119895 = 1 2 3) soTheorem C is a special case of Theorem 13

(ii) If 119896119895 = 1 (119895 = 1 2 119902) then 119902 = 5 if 119896119895 = 2 (119895 =1 2 119902) then 119902 = 4 if 119896119895 = 119897 ge 3 (119895 = 1 2 119902) then119902 = 3 So Theorem D is a special case of Theorem 13

In order to prove Theorem 13 we need the followinglemma

Lemma 15 (Li and Gu [13] see also Xuan [14]) Suppose thatΨ(119903) is a nonnegative increasing function in (1infin) and satisfies

lim sup119903rarrinfin

Ψ (119903)

log2119903= infin (60)

Then for any set 119864 sub (1infin) such that int119864(1119903 log 119903)119889119903 lt 13

one has

lim sup119903rarrinfin119903isin(1infin)minus119864

Ψ (119903)

log2119903= infin (61)

Proof Firstly we prove the following statement Let 119898 (119898 ge4) be a fixed positive integer 1205790 = 0 1205791 = 2120587119898 120579119898minus1 =(119898 minus 1)2120587119898 120579119898 = 1205790 We put Δ(120579119894) = 119911 | arg 119911 minus 120579119894| lt2120587119898 Δ119900(120579119894) = 119911 | arg 119911 minus 120579119894| lt 120587119898 119894 = 0 1 119898 minus 1Δ(120579119898) = Δ(1205790) Δ

119900(120579119898) = Δ119900(1205790) Then among these119898 angu-lar domains Δ(120579119894) there is at least an angular domain Δ(120579119894)such that for any system 119886119895 (119895 = 1 2 119902) of distinct valuesand any system 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positiveinteger or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (62)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (63)

Otherwise for any angular domainΔ(120579119894) (1 le 119894 le 119898) there isa system 119886

119895

119894(119895 = 1 2 119902) of distinct values and a system

119896119895

119894(119895 = 1 2 119902) such that 119896119895

119894is a positive integer or+infin and

that

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

) gt 2 (64)

for any 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

119873119896119895

119894) (119903 Δ (120579119894) 119886

119895

119894)

119879 (119903 119891)= 0 (65)

Put

119902

sum119895=1

(1 minus1

119896119895 + 1) = min1le119894le119898

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

)

gt 2 (66)

ApplyingTheorem 5 to Δ119900(120579119894+1) Δ(120579119894+1) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Δ119900 (120579119894+1) 119891)

le

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895

119894) (119903 Δ (120579119894+1) 119886

119895

119894+1)

+ 119874 (log2119903) + 120594 (119903 Δ (120579119894+1))

(67)

8 The Scientific World Journal

Noting119879(119903 119891) = sum119898minus1

119894=0119879(119903 Δ119900(120579119894+1) 119891) and adding two sides

of the above expression from 119894 = 0 to119898 minus 1 we can obtain

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 119891)

le119898minus1

sum119894=0

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895) (119903 Δ (120579119894+1) 119886119895

119894+1)

+ 119874 (log2119903) +119898minus1

sum119894=0

120594 (119903 Δ (120579119894+1))

(68)

For any 119894 there exists a 119903119894 the inequality 119879(119903 Δ119900(120579119894+1) 119891) gt

1198903119898 would bold for 119903 gt 119903119894 while the inequality (22) does notlook appropriate here Put 119864Δ119900(120579119894+1) is the set of 119903 which con-sists of a series of intervals and satisfies

int119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt

1

3119898 (69)

Let 1199030 = max119903119894 119894 = 1 2 119898 we have for any 119894 119879(1199030Δ119900(120579119894+1) 119891) gt 1198903119898 then

intcup119898minus1119894=0119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le119898minus1

sum119894=0

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt1

3

(70)

Applying Lemma 15 we have

lim sup119903rarrinfin119903isin(1infin)minus119864

119879 (119903 119891)

log2119903= infin (71)

where 119864 = cup119898minus1119894=0

119864Δ119900(120579119894+1) Therefore there exists a sequence1199031015840119899isin (1infin) minus 119864 such that

lim119899rarrinfin

119879 (1199031015840119899 119891)

log21199031015840119899

= infin (72)

It follows from (38) (68) and (72) that

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2) le 0 (73)

Hence119902

sum119895=1

(1 minus1

119896119895 + 1) le 2 (74)

This is a contradiction Hence for an arbitrary positive inte-ger 119898 there is at least an angular domain Δ(120579119894) such that forany system 119886119895 (119895 = 1 2 119902) of distinct values and any sys-tem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (75)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (76)

Choosing subsequence of 120579119898 still denote it 120579119898 we assumethat 120579119898 rarr 1205790 Put 119871 arg 119911 = 1205790 then 119871 is a pseudo-Tdirection that is stated in Definition 12

In fact for any 120576 (0 lt 120576 lt 1205872) when 119898 is sufficientlylarge we have Δ(120579119898) sub Ω(1205790 120576) By (76) we have

lim sup119903rarrinfin

119873119896119895) (119903 1205790 120576 119886119895)

119879 (119903 119891)ge lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119898) 119886119895)

119879 (119903 119891)gt 0

(77)

Hence Theorem 13 holds in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The first author was partly supported by the National NaturalScience Foundation of China (Grant no 11201395) and by theScience Foundation of Educational Commission of HubeiProvince (Grant no Q20132801) The second author waspartly supported by the NSF of Jiangxi Province (Grant20122BAB201006) The third author was partly supported byBeijing Natural Science Foundation (Grant no 1132013)

References

[1] J H Zheng ldquoOn transcendental meromorphic functions withradially distributed valuesrdquo Science in China A vol 47 no 3 pp401ndash416 2004

[2] L Yang Value Distribution Theory Springer Berlin Germany1993

[3] M Tsuji PotentialTheory inModern FunctionTheoryMaruzenTokyo Japan 1959

[4] D C Sun ldquoMain theorem on covering surfacesrdquo Acta Mathe-matica Scientia B vol 14 no 2 pp 213ndash225 1994

[5] D C Sun ldquoInequalities for covering surfaces and their applica-tionsrdquoChinese Annals ofMathematics A vol 18 no 1 pp 59ndash641997 (Chinese)

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs Clarendon Press Oxford UK 1964

[7] X L Zhang ldquoA fundamental inequality for meromorphic func-tions in an angular domain and its applicationrdquoActaMathemat-ica Sinica vol 10 no 3 pp 308ndash314 1994

[8] Q T Zhuang Singular Directions of Meromorphic Functionsvol 11 of Series of Monographs in Pure and AppliedMathematicsScience Press Beijing China 1982 (Chinese)

[9] C T Chuang ldquoOn Borel directions of meromorphic functionsof infinite order IIrdquoBulletin of theHongKongMathematical Soc-iety vol 2 no 2 pp 305ndash323 1999

[10] H Guo J H Zheng and TW Ng ldquoOn a new singular directionof meromorphic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 69 no 2 pp 277ndash287 2004

The Scientific World Journal 9

[11] Z-J Wu and D-C Sun ldquoOn the existence of 119879 direction ofmeromorphic function concerning multiple valuesrdquo KodaiMathematical Journal vol 31 no 1 pp 133ndash149 2008

[12] J H Zheng Value Distribution of Meromorphic FunctionsTsinghua University Press Beijing China 2010

[13] C H Li and Y X Gu ldquoA fundamental inequality for 119870-quasimeromorphic mappings in an angular domain and itsapplicationrdquo Acta Mathematica Sinica vol 49 no 6 pp 1279ndash1286 2006 (Chinese)

[14] Z-X Xuan ldquoOn the existence of 119879-direction of algebroidfunctions a problem of J H Zhengrdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 540ndash547 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

Theorem C was conjectured by Zheng [1] In [11] theauthors study the existence of T direction of 119891(119911) concerningmultiple values We call 119869 arg 119911 = 120579 the T direction of 119891(119911)concerningmultiple values provided that given any 119886 isin Cinfinpossibly with exception of at most [(2119897 + 2)119897] values of 119886 forany positive number 120576 lt 120587 we have

lim sup119903rarrinfin

119873119897) (119903 120579 120576 119886)

119879 (119903 119891)gt 0 (57)

where [119909] implies the maximum integer number which doesnot exceed 119909 and 119897 is a positive integer

Theorem D Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a T direction of 119891(119911) concerningmultiple values

Note that the T direction of meromorphic function con-cerning multiple values is a refinement of the ordinary T dir-ection since [(2119897+2)119897] rarr 2 as 119897 rarr infin Since Zheng [1] gavethe definition of T direction then there is a considerablenumber result related this directionwe refer the reader to [12]for finding a careful discussion of this direction

It is well known that T direction is a concept in angulardistribution which corresponds to the Nevanlinna secondfundamental theorem in module distribution It is natural toconsider the corresponding result to Theorem B in angulardistribution

Definition 12 Let 119891(119911) be a meromorphic function A direc-tion arg 119911 = 120579 is called a pseudo-T direction of119891(119911) if for anynumber 120576 (0 lt 120576 lt 1205872) any system 119886119895 (119895 = 1 2 119902) ofdistinct values and any system 119896119895 (119895 = 1 2 119902) such that 119896119895is a positive integer or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (58)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 120579 120576 119886119895)

119879 (119903 119891)gt 0 (59)

Theorem 13 Let 119891(119911) be a meromorphic function and satisfy(56) Then there at least exists a pseudo-T direction of 119891(119911)

Remark 14 (i) In Theorem C 119902 = 3 119896119895 = infin (119895 = 1 2 3) soTheorem C is a special case of Theorem 13

(ii) If 119896119895 = 1 (119895 = 1 2 119902) then 119902 = 5 if 119896119895 = 2 (119895 =1 2 119902) then 119902 = 4 if 119896119895 = 119897 ge 3 (119895 = 1 2 119902) then119902 = 3 So Theorem D is a special case of Theorem 13

In order to prove Theorem 13 we need the followinglemma

Lemma 15 (Li and Gu [13] see also Xuan [14]) Suppose thatΨ(119903) is a nonnegative increasing function in (1infin) and satisfies

lim sup119903rarrinfin

Ψ (119903)

log2119903= infin (60)

Then for any set 119864 sub (1infin) such that int119864(1119903 log 119903)119889119903 lt 13

one has

lim sup119903rarrinfin119903isin(1infin)minus119864

Ψ (119903)

log2119903= infin (61)

Proof Firstly we prove the following statement Let 119898 (119898 ge4) be a fixed positive integer 1205790 = 0 1205791 = 2120587119898 120579119898minus1 =(119898 minus 1)2120587119898 120579119898 = 1205790 We put Δ(120579119894) = 119911 | arg 119911 minus 120579119894| lt2120587119898 Δ119900(120579119894) = 119911 | arg 119911 minus 120579119894| lt 120587119898 119894 = 0 1 119898 minus 1Δ(120579119898) = Δ(1205790) Δ

119900(120579119898) = Δ119900(1205790) Then among these119898 angu-lar domains Δ(120579119894) there is at least an angular domain Δ(120579119894)such that for any system 119886119895 (119895 = 1 2 119902) of distinct valuesand any system 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positiveinteger or +infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (62)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (63)

Otherwise for any angular domainΔ(120579119894) (1 le 119894 le 119898) there isa system 119886

119895

119894(119895 = 1 2 119902) of distinct values and a system

119896119895

119894(119895 = 1 2 119902) such that 119896119895

119894is a positive integer or+infin and

that

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

) gt 2 (64)

for any 119895 (1 le 119895 le 119902) we have

lim sup119903rarrinfin

119873119896119895

119894) (119903 Δ (120579119894) 119886

119895

119894)

119879 (119903 119891)= 0 (65)

Put

119902

sum119895=1

(1 minus1

119896119895 + 1) = min1le119894le119898

119902

sum119895=1

(1 minus1

119896119895

119894+ 1

)

gt 2 (66)

ApplyingTheorem 5 to Δ119900(120579119894+1) Δ(120579119894+1) we have

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 Δ119900 (120579119894+1) 119891)

le

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895

119894) (119903 Δ (120579119894+1) 119886

119895

119894+1)

+ 119874 (log2119903) + 120594 (119903 Δ (120579119894+1))

(67)

8 The Scientific World Journal

Noting119879(119903 119891) = sum119898minus1

119894=0119879(119903 Δ119900(120579119894+1) 119891) and adding two sides

of the above expression from 119894 = 0 to119898 minus 1 we can obtain

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 119891)

le119898minus1

sum119894=0

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895) (119903 Δ (120579119894+1) 119886119895

119894+1)

+ 119874 (log2119903) +119898minus1

sum119894=0

120594 (119903 Δ (120579119894+1))

(68)

For any 119894 there exists a 119903119894 the inequality 119879(119903 Δ119900(120579119894+1) 119891) gt

1198903119898 would bold for 119903 gt 119903119894 while the inequality (22) does notlook appropriate here Put 119864Δ119900(120579119894+1) is the set of 119903 which con-sists of a series of intervals and satisfies

int119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt

1

3119898 (69)

Let 1199030 = max119903119894 119894 = 1 2 119898 we have for any 119894 119879(1199030Δ119900(120579119894+1) 119891) gt 1198903119898 then

intcup119898minus1119894=0119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le119898minus1

sum119894=0

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt1

3

(70)

Applying Lemma 15 we have

lim sup119903rarrinfin119903isin(1infin)minus119864

119879 (119903 119891)

log2119903= infin (71)

where 119864 = cup119898minus1119894=0

119864Δ119900(120579119894+1) Therefore there exists a sequence1199031015840119899isin (1infin) minus 119864 such that

lim119899rarrinfin

119879 (1199031015840119899 119891)

log21199031015840119899

= infin (72)

It follows from (38) (68) and (72) that

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2) le 0 (73)

Hence119902

sum119895=1

(1 minus1

119896119895 + 1) le 2 (74)

This is a contradiction Hence for an arbitrary positive inte-ger 119898 there is at least an angular domain Δ(120579119894) such that forany system 119886119895 (119895 = 1 2 119902) of distinct values and any sys-tem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (75)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (76)

Choosing subsequence of 120579119898 still denote it 120579119898 we assumethat 120579119898 rarr 1205790 Put 119871 arg 119911 = 1205790 then 119871 is a pseudo-Tdirection that is stated in Definition 12

In fact for any 120576 (0 lt 120576 lt 1205872) when 119898 is sufficientlylarge we have Δ(120579119898) sub Ω(1205790 120576) By (76) we have

lim sup119903rarrinfin

119873119896119895) (119903 1205790 120576 119886119895)

119879 (119903 119891)ge lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119898) 119886119895)

119879 (119903 119891)gt 0

(77)

Hence Theorem 13 holds in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The first author was partly supported by the National NaturalScience Foundation of China (Grant no 11201395) and by theScience Foundation of Educational Commission of HubeiProvince (Grant no Q20132801) The second author waspartly supported by the NSF of Jiangxi Province (Grant20122BAB201006) The third author was partly supported byBeijing Natural Science Foundation (Grant no 1132013)

References

[1] J H Zheng ldquoOn transcendental meromorphic functions withradially distributed valuesrdquo Science in China A vol 47 no 3 pp401ndash416 2004

[2] L Yang Value Distribution Theory Springer Berlin Germany1993

[3] M Tsuji PotentialTheory inModern FunctionTheoryMaruzenTokyo Japan 1959

[4] D C Sun ldquoMain theorem on covering surfacesrdquo Acta Mathe-matica Scientia B vol 14 no 2 pp 213ndash225 1994

[5] D C Sun ldquoInequalities for covering surfaces and their applica-tionsrdquoChinese Annals ofMathematics A vol 18 no 1 pp 59ndash641997 (Chinese)

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs Clarendon Press Oxford UK 1964

[7] X L Zhang ldquoA fundamental inequality for meromorphic func-tions in an angular domain and its applicationrdquoActaMathemat-ica Sinica vol 10 no 3 pp 308ndash314 1994

[8] Q T Zhuang Singular Directions of Meromorphic Functionsvol 11 of Series of Monographs in Pure and AppliedMathematicsScience Press Beijing China 1982 (Chinese)

[9] C T Chuang ldquoOn Borel directions of meromorphic functionsof infinite order IIrdquoBulletin of theHongKongMathematical Soc-iety vol 2 no 2 pp 305ndash323 1999

[10] H Guo J H Zheng and TW Ng ldquoOn a new singular directionof meromorphic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 69 no 2 pp 277ndash287 2004

The Scientific World Journal 9

[11] Z-J Wu and D-C Sun ldquoOn the existence of 119879 direction ofmeromorphic function concerning multiple valuesrdquo KodaiMathematical Journal vol 31 no 1 pp 133ndash149 2008

[12] J H Zheng Value Distribution of Meromorphic FunctionsTsinghua University Press Beijing China 2010

[13] C H Li and Y X Gu ldquoA fundamental inequality for 119870-quasimeromorphic mappings in an angular domain and itsapplicationrdquo Acta Mathematica Sinica vol 49 no 6 pp 1279ndash1286 2006 (Chinese)

[14] Z-X Xuan ldquoOn the existence of 119879-direction of algebroidfunctions a problem of J H Zhengrdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 540ndash547 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

Noting119879(119903 119891) = sum119898minus1

119894=0119879(119903 Δ119900(120579119894+1) 119891) and adding two sides

of the above expression from 119894 = 0 to119898 minus 1 we can obtain

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2)119879 (119903 119891)

le119898minus1

sum119894=0

119902

sum119895=1

119896119895

119894

119896119895

119894+ 1

119873119896119895) (119903 Δ (120579119894+1) 119886119895

119894+1)

+ 119874 (log2119903) +119898minus1

sum119894=0

120594 (119903 Δ (120579119894+1))

(68)

For any 119894 there exists a 119903119894 the inequality 119879(119903 Δ119900(120579119894+1) 119891) gt

1198903119898 would bold for 119903 gt 119903119894 while the inequality (22) does notlook appropriate here Put 119864Δ119900(120579119894+1) is the set of 119903 which con-sists of a series of intervals and satisfies

int119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt

1

3119898 (69)

Let 1199030 = max119903119894 119894 = 1 2 119898 we have for any 119894 119879(1199030Δ119900(120579119894+1) 119891) gt 1198903119898 then

intcup119898minus1119894=0119864Δ119900(120579119894+1)

1

119903 log 119903119889119903 le119898minus1

sum119894=0

1

log119879 (119903 Δ119900 (120579119894+1) 119891)lt1

3

(70)

Applying Lemma 15 we have

lim sup119903rarrinfin119903isin(1infin)minus119864

119879 (119903 119891)

log2119903= infin (71)

where 119864 = cup119898minus1119894=0

119864Δ119900(120579119894+1) Therefore there exists a sequence1199031015840119899isin (1infin) minus 119864 such that

lim119899rarrinfin

119879 (1199031015840119899 119891)

log21199031015840119899

= infin (72)

It follows from (38) (68) and (72) that

(

119902

sum119895=1

(1 minus1

119896119895 + 1) minus 2) le 0 (73)

Hence119902

sum119895=1

(1 minus1

119896119895 + 1) le 2 (74)

This is a contradiction Hence for an arbitrary positive inte-ger 119898 there is at least an angular domain Δ(120579119894) such that forany system 119886119895 (119895 = 1 2 119902) of distinct values and any sys-tem 119896119895 (119895 = 1 2 119902) such that 119896119895 is a positive integer or+infin and that

119902

sum119895=1

(1 minus1

119896119895 + 1) gt 2 (75)

there exists at least one integer 119895 (1 le 119895 le 119902) such that

lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119894) 119886119895)

119879 (119903 119891)gt 0 (76)

Choosing subsequence of 120579119898 still denote it 120579119898 we assumethat 120579119898 rarr 1205790 Put 119871 arg 119911 = 1205790 then 119871 is a pseudo-Tdirection that is stated in Definition 12

In fact for any 120576 (0 lt 120576 lt 1205872) when 119898 is sufficientlylarge we have Δ(120579119898) sub Ω(1205790 120576) By (76) we have

lim sup119903rarrinfin

119873119896119895) (119903 1205790 120576 119886119895)

119879 (119903 119891)ge lim sup119903rarrinfin

119873119896119895) (119903 Δ (120579119898) 119886119895)

119879 (119903 119891)gt 0

(77)

Hence Theorem 13 holds in this case

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The first author was partly supported by the National NaturalScience Foundation of China (Grant no 11201395) and by theScience Foundation of Educational Commission of HubeiProvince (Grant no Q20132801) The second author waspartly supported by the NSF of Jiangxi Province (Grant20122BAB201006) The third author was partly supported byBeijing Natural Science Foundation (Grant no 1132013)

References

[1] J H Zheng ldquoOn transcendental meromorphic functions withradially distributed valuesrdquo Science in China A vol 47 no 3 pp401ndash416 2004

[2] L Yang Value Distribution Theory Springer Berlin Germany1993

[3] M Tsuji PotentialTheory inModern FunctionTheoryMaruzenTokyo Japan 1959

[4] D C Sun ldquoMain theorem on covering surfacesrdquo Acta Mathe-matica Scientia B vol 14 no 2 pp 213ndash225 1994

[5] D C Sun ldquoInequalities for covering surfaces and their applica-tionsrdquoChinese Annals ofMathematics A vol 18 no 1 pp 59ndash641997 (Chinese)

[6] W K Hayman Meromorphic Functions Oxford MathematicalMonographs Clarendon Press Oxford UK 1964

[7] X L Zhang ldquoA fundamental inequality for meromorphic func-tions in an angular domain and its applicationrdquoActaMathemat-ica Sinica vol 10 no 3 pp 308ndash314 1994

[8] Q T Zhuang Singular Directions of Meromorphic Functionsvol 11 of Series of Monographs in Pure and AppliedMathematicsScience Press Beijing China 1982 (Chinese)

[9] C T Chuang ldquoOn Borel directions of meromorphic functionsof infinite order IIrdquoBulletin of theHongKongMathematical Soc-iety vol 2 no 2 pp 305ndash323 1999

[10] H Guo J H Zheng and TW Ng ldquoOn a new singular directionof meromorphic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 69 no 2 pp 277ndash287 2004

The Scientific World Journal 9

[11] Z-J Wu and D-C Sun ldquoOn the existence of 119879 direction ofmeromorphic function concerning multiple valuesrdquo KodaiMathematical Journal vol 31 no 1 pp 133ndash149 2008

[12] J H Zheng Value Distribution of Meromorphic FunctionsTsinghua University Press Beijing China 2010

[13] C H Li and Y X Gu ldquoA fundamental inequality for 119870-quasimeromorphic mappings in an angular domain and itsapplicationrdquo Acta Mathematica Sinica vol 49 no 6 pp 1279ndash1286 2006 (Chinese)

[14] Z-X Xuan ldquoOn the existence of 119879-direction of algebroidfunctions a problem of J H Zhengrdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 540ndash547 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 9

[11] Z-J Wu and D-C Sun ldquoOn the existence of 119879 direction ofmeromorphic function concerning multiple valuesrdquo KodaiMathematical Journal vol 31 no 1 pp 133ndash149 2008

[12] J H Zheng Value Distribution of Meromorphic FunctionsTsinghua University Press Beijing China 2010

[13] C H Li and Y X Gu ldquoA fundamental inequality for 119870-quasimeromorphic mappings in an angular domain and itsapplicationrdquo Acta Mathematica Sinica vol 49 no 6 pp 1279ndash1286 2006 (Chinese)

[14] Z-X Xuan ldquoOn the existence of 119879-direction of algebroidfunctions a problem of J H Zhengrdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 540ndash547 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of