Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 426302, 11 pages
doi:10.1155/2012/426302
Research Article
The Filling Discs Dealing with Multiple Values of
an Algebroid Function in the Unit Disc
Zu-Xing Xuan1 and Nan Wu2
1
Beijing Key Laboratory of Information Service Engineering, Department of General Education, Beijing
Union University, No. 97 Bei Si Huan Dong Road, Chaoyang District, Beijing 100101, China
2
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Correspondence should be addressed to Zu-Xing Xuan, xuanzuxing@ss.buaa.edu.cn
Received 7 August 2011; Accepted 18 December 2011
Academic Editor: Gabriel Turinici
Copyright q 2012 Z.-X. Xuan and N. Wu. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
In this paper, by using the potential theory we prove the existence of filling discs dealing with
multiple values of an algebroid function of finite order defined in the unit disc.
1. Introduction and Main Result
The value distribution theory of meromorphic functions due to Hayman see 1 for standard
references was extended to the corresponding theory of algebroid functions by Selbreg 2,
Ullrich 3, and Valiron 4 around 1930. The filling discs of an algebroid function are an
important part of the value distribution theory. For an algebroid function defined on z-plane,
the existence of its filling discs was proved by Sun 5 in 1995. In 1997, for the algebroid
functions of infinite order and zero order, Gao 6 obtained the the corresponding results. The
existence of the sequence of filling discs of algebroid functions dealing with multiple values,
of finite or infinite order, was first proved by Gao 7, 8. The existence of filling discs in the
strong Borel direction of algebroid function with finite order was proved by Huo and Kong
in 9. Compared with the case of C, it is interesting to investigate the algebroid functions
defined in the unit disc, and there are some essential differences between these two cases.
Recently, the first author 10 has investigated this problem and confirmed the existence of
filling discs for this case. In this note, we will continue the work of Xuan 10 by considering
the case dealing with multiple values and get more precise results.
Let w wz z ∈ Δ be the ν-valued algebroid function defined by irreducible
equation
Aν zwν Aν−1 zwν−1 · · · A0 z 0,
1.1
2
Abstract and Applied Analysis
where Aν z, . . . , A0 z are entire functions without any common zeros. The single-valued
domain of definition of wz is a ν-valued covering of the z-plane, a Riemann surface,
z , whose projection in the z-plane is z, is denoted by z. The
z . A point in R
denoted by R
z| < r.
part of Rz , which covers the disc {z : |z| < r}, is denoted by |
Denote
1
Sr, w π
|w z|
2
1 |wz|2
|
z|≤r
dω.
1.2
Sr, w is called the mean covering number of |
z| ≤ r into w-sphere under the mapping
w wz. And Sr, w is conformal invariant. Let nr, a be the number of zeros of wz − a,
counted according to their multiplicities in |
z| ≤ r. nl E, w α denotes the number of zeros
with multiplicity ≤ l of wz α in E, each zero being counted only once.
Let
r
n0, a
nt, a − n0, a
dt log r,
t
ν
0
ν
1
1
mr, a log iθ dθ, z reiθ ,
wj re − a 2πν |z|r j1
Nr, a 1
ν
1.3
where |
z| r is the boundary of |
z| ≤ r. The characteristic function of wz is defined by
T r, w 1
ν
r
0
St, w
dt.
t
1.4
In view of 4, we have
T r, w mr, w Nr, ∞ O1.
1.5
The order of algebroid function wz is defined by
ρ lim sup
r → 1−
log T r, w
.
log1/1 − r
1.6
In this paper we assume that 0 < ρ < ∞, V is the w-sphere, and C is a constant
z be the number of the branch points of R
z
which can stand for different constant. Let nr, R
in |
z| ≤ r, counted with the order of branch. Write
z
N r, R
r n t, R
z
z − n 0, R
z
n
0,
R
1
dt log r.
ν 0
t
ν
1.7
Valiron is the first one to introduce the concept of a proximate order ρ1/1 − r for a
meromorphic function wz with finite positive order and U1/1 − r 1/1 − rρ1/1−r
Abstract and Applied Analysis
3
is called type function of wz or T r, w such that ρ1/1 − r is nondecreasing, piecewise
continuous, and differentiable, and
lim ρ
r → 1−
Uk/1 − r
kρ
r → 1− U1/1 − r
lim
1
1−r
ρ,
k is any given positive constant ,
T r, w
1,
r → 1− U1/1 − r
1/1 − rρ−ε
0, 0 < ε < ρ.
lim
r → 1− U1/1 − r
1.8
lim
For an algebroid function wz of finite positive order, we can apply the same method
to get its type function U1/1 − r.
Our main result is the following.
Theorem 1.1. Suppose that wz is the ν-valued algebroid function of finite order ρ in |z| < 1 defined
by 1.1 and l(≥ 2ν 1) is an integer, then there exists a sequence of discs
Γn : {|z − zn | < rn σn },
n 1, 2, . . . ,
1.9
where
zn rn eiθn ,
lim rn 1,
n→∞
σn > 0,
lim σn 0.
n→∞
1.10
Such that for each α
nl Γn ∩ , w α ≥
1
1 − rn ρ1−εn
,
1.11
except for those complex numbers contained in the union of 2ν spherical discs each with radius 1 −
rn ρ/11 , where limn → ∞ εn 0, Δ {z : |z| < 1}.
The discs with the above property are called filling discs dealing with multiple values.
Remark 1.2. In 10, the result says that nΓn ∩ , w α ≥ 1/1 − rn ρ−εn . Theorem 1.1 is really
the improvement of 10.
Remark 1.3. The existence of filling discs in Borel radius of meromorphic functions was
proved by Kong 11. In view of our theorem, we can get the similar results of 11 when
ν 1. But we must point out that the structure and definition of filling discs between this
paper and 11 are different. There are also some papers relevant to the singular points of
algebroid functions in the unit disc see 12–14.
4
Abstract and Applied Analysis
2. Two Lemmas
Lemma 2.1 see 15 or 16. Suppose that wz is the ν-valued algebroid function in {z : |z| < R}
defined by 1.1, l≥ 2ν 1 is an integer and a1 , a2 , a3 , . . . , aq q ≥ 3 are distinct points given
arbitrarily in w-sphere, and the spherical distance of any two points is no smaller than δ ∈ 0, 1/2,
then for any r ∈ 0, R, one has
q
l1 CR
2
z n R, R
nl R, aj .
q−2−
Sr, w ≤
l
l
−
rδ10
R
j1
2.1
Combining the potential theory with Lemma 2.1, one proves Lemma 2.2, which is
crucial to the theorem.
Lemma 2.2. Suppose that wz is the ν-valued algebroid function of finite order ρ satisfying 0 < ρ <
∞ in |z| < 1 defined by 1.1 and l≥ 2ν 1 is an integer. For any ε ∈ 0, ρ, 0 < R < 1, there
exists a0 ∈ 1/2, 1, such that for any a ∈ a0 , 1, put
2π q 1
2π
,
θq ,
1−a
m
2π
1 − ap ≤ |z| < 1 − ap2 ∩ arg z − θq ≤
p 1, 2, 3, . . . ; q 0, 1, 2, . . . , m − 1 ,
m
2.2
rn 1 − an ,
Ωpq
m
where x stands for the inter part of x.
Then, among p, q, there exists at least one pair p0 , q0 , such that 1 − ap0 > R, and in Ωp0 q0 ,
nl Ωp0 q0 , w α ≥
1
aρ1−ε
2.3
,
except for those complex numbers contained in the union of 2ν spherical discs each with radius δ ap0 ρ/11 .
Proof. Suppose the conclusion is false. Then there exists a sequence {ai }∞
i1 0 < ai < 1, where
limi → ∞ ai 1. For any a ∈ {ai }, any p > P log1 − R/ log a and q ∈ {0, 1, 2, . . . , m − 1}, there
exist 2ν 1 complex numbers which satisfy that the spherical distance of any two of those
points is no smaller than δ apρ/40 . Denote
2ν1
αj αj p, q j1 .
2.4
For any p, q mentioned above, we have
nl Ωpq , w αj <
1
apρ1−ε
.
For any r > R, let T log1 − r/ log a, then we have 1 − aT ≤ r < 1 − aT 1 .
2.5
Abstract and Applied Analysis
5
For any given positive integers N and M, set
b a1/M ∈ 0, 1,
γpt 1 − bMpt , t 0, 1, 2, . . . , M − 1,
Lpt γpt ≤ |z| < γp,t1 ,
2πq 2πj
,
θqj m
Nm
qj z : |z| < 1 − aT , θqj ≤ arg z < θq,j1 .
2.6
Then
−1
M−1
T
1 − a ≤ |z| < 1 − aT Lpt ,
t0 p1
|z| < 1 − aT N−1
m−1
2.7
qj .
j0 q0
Thus there exists t0 , j0 which are related to T . We can assume t0 0, j0 0, such that
T −1 z ≤ 1 n 1 − aT , R
z ,
n Lp0 , R
M
p1
z ≤ 1 n 1 − aT , R
z .
n q0 , R
N
q0
m−1
2.8
Set
θq1,0 θq1,1
θq0 θq1
bMpM bMpM1
bMp bMp1
≤ |z| < 1 −
∩
≤ arg z <
,
1−
2
2
2
2
Ωpq 1 − bMp ≤ |z| < 1 − bMpM1 ∩ θq0 ≤ arg z < θq1,1 .
Ω0pq
2.9
Then we have
Ω0pq ⊂ Ωpq ⊂ Ωpq .
Since {Ωpq }p,q covers
T −1
p1
Lp0 and
m−1
q0
2.10
q0 twice at most. We obtain
T −1m−1
z ≤ 1 1 1 n 1 − aT , R
z .
n Ωpq , R
M N
p1 q0
2.11
Obviously, each Ωpq can be mapped conformally to the unit disc |ζ| < 1 such that the center
of Ωpq is mapped to ζ 0, and the image of Ω0pq is contained in the disc |ζ| < η<1. Since
6
Abstract and Applied Analysis
all Ωpq , Ω0pq are similar, C is independent of p, q. Since S is conformally invariant, in view of
Lemma 2.1, we obtain
T −1 m−1
2 2 2 T
0
S Ωpq , w 2ν − 1 −
2ν − 1 −
S 1 − a , w ≤ 2ν − 1 −
S 1 − a2 , w
l
l pP 1 q0
l
⎤
⎡
T −1 m−1
l1 2ν1
l C
z ⎣
n Ωpq , R
≤
n Ωpq , w αj ⎦
10 1 − η
l
δ
j1
pP 1 q0
2 2ν − 1 −
S 1 − a2 , w
l
1
l1
1
−T ρ1−ε
z
≤ 3νT ma
1
n 1 − aT , R
l
M N
10 2 S 1 − a2 , w .
2ν − 1 −
CT a−T ρ/11
l
2.12
For sufficiently large integer T log1 − r/ log a, r ∈ 1 − aT , 1 − aT 1 . Thus we get
1−r
2 2
S 1−
, w ≤ 2ν − 1 −
S 1 − aT , w
2ν − 1 −
l
a
l
1
1
l1
1
z
1
n
r,
R
≤
l
M N
1 − rρ1−ε/2
10ρ/11
1
C,
C
1−r
2.13
where C is a constant.
For any integer T log1 − r/ log a, we have 1 − aT /a − aT ∈ 1/a, 1 a/a.
We can choose one a> l 1/l2ν − 2/2ν − 1 − 2/l ∈ 0, 1 as l ≥ 2ν 1 such that
l/l 12ν − 1 − 2/l/2ν − 2 ∈ 1/a, 1 a/a. For a certain sufficiently fixed large
inter T , we have
−1
2ν − 1 − 2/l
1 − aT
l
·
.
<
l1
2ν − 2
a − aT
2.14
1 − 1 − t/a 1
1
1
−
−1
t
a
a
t
1
1
1
1
1
1
−1
1
−
,
≥ −
T
T
−1
a
a
a
ak
1−a
1 a ··· a
2.15
k
1
1−
1 a a2 · · · aT −1
This yields the following:
where t ∈ 1 − aT , 1 − aT 1 .
Abstract and Applied Analysis
7
Hence
r
1−aT
r
S1 − 1 − t/a, w 1 − 1 − t/a
dt
1 − 1 − t/a
t
r
1
S1 − 1 − t/a, w
≥
dt
ak 1−aT
1 − 1 − t/a
1−1−r/a
1
Sx, w
dx.
k 1−aT −1
x
S1 − 1 − t/a, w
dt t
1−aT
2.16
Next, we deduce the following:
r
1
1−aT
ρ1−ε/2
1 − t
t
dt ≤
1
1 − aT
−
≤
r
1
1 − aT
1
1−aT
r
1 − t
1−aT
ρ1−ε/2
dt
1
1 − t
ρ1−ε/2
d1 − t
2.17
1
1
1
·
·
,
T
ρ
−
ε/2
1−a
1 − rρ−ε/2
when r ≥ t ≥ 1 − aT .
In view of 1 − t ≤ t for t ∈ 1 − aT , 1 − aT 1 , we have
r
1
1−aT
10ρ/11
1 − t
t
dt ≤
r
1
1−aT
1 − t
10ρ/111
dt ≤
1
1
·
.
10ρ/11 1 − r10ρ/11
2.18
Dividing both sides of 2.13 by νt and integrating it from 1 − aT to r, we have
2 1 r S1 − 1 − t/a, w
dt
2ν − 1 −
l ν 1−aT
t
1 r
1
1
1
l1
≤
1
dt
ν 1−aT 1 − tρ1−ε/2 t
l
M N
⎤
⎡
r n t, R
z
z − n 0, R
z
z
n
0,
R
n
0,
R
⎥
⎢1
dt log r −
log 1 − aT ⎦
×⎣
ν 1−aT
t
ν
ν
C
ν
r
1−aT
1
1 − t
10ρ/11
t
dt ≤
⎡
1
1 ⎢1
l1
· 1
⎣
l
M N
ν
1
1
1
1
·
·
·
T
ν 1 − a ρ − ε/2 1 − rρ−ε/2
⎤
r n t, R
z
z − n 0, R
z
n 0, R
⎥
dt log r ⎦
t
ν
0
C
1
1 n 0, Rz
1
1
l1
· 1
log 1 − aT −
l
M N
ν
ν 10ρ/11 1 − r10ρ/11
8
Abstract and Applied Analysis
1
1
1
1
l1
1
z
·
·
1
N
r,
R
·
ν ρ − ε/2 1 − rρ−ε/2
l
M N
C
1
1 n 0, Rz
1
1
l1
· 1
log 1 − aT .
−
l
M N
ν
ν 10ρ/11 1 − r10ρ/11
2.19
Note that T is fixed, we see that T 1 − aT −1 , w is a finite constant.
Hence,
2 1 1 1−1−r/a St, w
dt
2ν − 1 −
l k ν 1−aT −1
t
2 1 r S1 − 1 − t/a
≤ 2ν − 1 −
dt
l ν 1−aT
t
1
1
1
1
1
1
l1
z
·
1
N
r,
R
≤
·
l
M N
1 − aT ν ρ − ε/2 1 − rρ−ε/2
C
l1
1
1 n 0, Rz
1
1
−
log 1 − aT 1
.
l
M N
ν
ν 10ρ/11 1 − r10ρ/11
2.20
Then
2 1 1−1−r/a St, w
1
1
1
1
2ν − 1 −
dt ≤
·
l kν 0
t
1 − aT ν ρ − ε/2 1 − rρ−ε/2
n 0, R
z
l 1
1
1
1
1
l1
2.21
z −
log 1 − aT
1
N r, R
1
l
M N
l
M N
ν
1
1
C
1 T −1
T
1
−
a
,
w
.
ν 10ρ/11 1 − r10ρ/11 k
In view of 3, we know that
z ≤ 2ν − 1T r, w O1.
N r, R
2.22
We obtain
2
2ν − 1 −
l
1−r
1
1
1
1
,w ≤
· · T 1−
k
a
1 − aT ν ρ − ε/2 1 − rρ−ε/2
1
1
l1
· 1
2ν − 1T r, w
l
M N
C
1
1
C log 1 − aT C,
ν 10ρ/11 1 − r10ρ/11
where C is a constant.
2.23
Abstract and Applied Analysis
9
Dividing both sides of the above inequality by U1/1 − r 1/1 − rρ1/1−r , we have
1
1
1 T 1 − 1 − r/a, w
1 − rρ1/1−r
≤
·
k
U1/1 − r
1 − aT ν ρ − ε/2 1 − rρ−ε/2
T r, w
1
1
l1
· 1
2ν − 1
l
M N
U1/1 − r
log 1 − aT
1
C
C
1 − rρ1/1−r
C
.
10ρ/11
ρ1/1−r
ν 10ρ/11 1 − r
1/1 − r
1/1 − rρ1/1−r
2ν − 1 −
2
l
·
2.24
We note that
T 1 − 1 − r/a, w T 1 − 1 − r/a, w Ua/1 − r
.
U1/1 − r
Ua/1 − r
U1/1 − r
2.25
In view of the properties of the U1/1 − r, we obtain
lim sup
r → 1−
T 1 − 1 − r/a, w
T 1 − 1 − r/a, w
Ua/1 − r
≥ lim sup
lim inf
r
→
1−
U1/1 − r
Ua/1
−
r
U1/1 − r
r → 1−
T 1 − 1 − r/a, w
Ua/1 − r
lim
aρ .
lim sup
r
→
1−
Ua/1
−
r
U1/1 − r
r → 1−
2.26
Letting r → 1− in 2.24, we have
2ν − 1 −
1
2 1 ρ l1
1
a ≤
1
2ν − 1,
l k
l
M N
2.27
2ν − 1 −
1
1
2 l1
≤
1
2ν − 1ka−ρ .
l
l
M N
2.28
that is,
Letting a → 1−, M → ∞, and N → ∞, we obtain
k≥
l 2ν − 1 − 2/l
.
l1
2ν − 2
2.29
This is contradictory to k < l/l 12ν − 1 − 2/l/2ν − 2, and the lemma is proved.
3. Proof of the Theorem
Proof. Choose εn ρ/2n, Rn 1 − 1/2n .
10
Abstract and Applied Analysis
In view of Lemma 2.2, there exists an ∈ 1 − 1/n, 1, mn 2π/1 − an , pn , qn , θqn 2πqn 1/mn , and
p
p 2
Ωpn qn 1 − ann ≤ |z| ≤ 1 − ann
arg z − θq ≤ 2π ,
n
mn
n 1, 2, . . ..
3.1
Let
p
zn 1 − ann eiθn .
θn θqn ,
3.2
Then
p
1 > rn |zn | 1 − ann > Rn 1 −
1
−→ 1 − n −→ ∞,
2n
p
lim ann 0.
n → ∞
3.3
Set
Bn ≤
!
p 2
1 − ann
!
2pn
1 − an
1−
p
ann
" p
p 2 2π
− 1 − ann 1 − ann
mn
" 2π
p
2p
− 1 − ann 1 − an n
mn
p
ann
1−
p
ann
1
p
ann
p 2π p
p
≤ 1 − ann ann 1 ann
.
mn
2π
3.4
mn
Take
2π
p
p
,
σn ann 1 ann
mn
3.5
then
σn −→ 0
n −→ ∞.
3.6
Γn {|z − zn | < rn σn }.
3.7
Ωpn qn ⊂ Γn .
3.8
Put
Then
Abstract and Applied Analysis
11
In view of Lemma 2.2, for each n,
nl Γn ∩ , w α ≥
1
ρ1−ε
an n
,
3.9
except for those complex numbers contained in the union of 2ν spherical discs each with rap ρ/11
1 − rn ρ/11 . Theorem 1.1 is proved.
dius ann
Remark 3.1. By using the same method, we can prove the existence of filling discs for Kquasimeromorphic mappings whose general case is carefully discussed in another paper.
Acknowledgments
The authors would like to thank the referee for his/her many helpful suggestions on an
early version of the paper. Z.-X. Zuxing is supported in part by Beijing Municipal Research
Foundation for The Excellent Scholars Ministry 2011D005022000009; Science and Technology Research Program of Beijng Municipal Commission of Education KM201211417011;
Funding Project for Academic Human Resources Development in Beijing Union University.
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World Journal
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Volume 2014
International Journal of
Differential Equations
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Volume 2014
Volume 2014
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International Journal of
Advances in
Combinatorics
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Mathematical Physics
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Volume 2014
Journal of
Complex Analysis
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Volume 2014
International
Journal of
Mathematics and
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Sciences
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Stochastic Analysis
Abstract and
Applied Analysis
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International Journal of
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Volume 2014
Volume 2014
Discrete Dynamics in
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Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
Optimization
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Volume 2014
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Volume 2014