Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 518972, 13 pages
doi:10.1155/2011/518972
Research Article
An Inequality of Meromorphic Vector Functions
and Its Application
Wu Zhaojun1, 2 and Chen Yuxian2
1
2
School of Mathematics and Statistics, Xianning University, Xianning 437100, China
School of Mathematics and Computer Science, Xinyu University, Xinyu 338004, China
Correspondence should be addressed to Wu Zhaojun, wuzj52@hotmail.com
Received 17 July 2011; Accepted 7 November 2011
Academic Editor: Gerd Teschke
Copyright q 2011 W. Zhaojun and C. Yuxian. 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.
Firstly, an inequality for vector-valued meromorphic functions is established which extend a
corresponding inequality of Milloux for meromorphic scalar-valued function 1946. As an
application, the relationship between the characteristic function of a vector-valued meromorphic
function f and its derivative f is studied, results are obtained to extend some related results for
meromorphic scalar-valued function of Weitsman 1969 and Singh and Gopalakrishna 1971.
1. Introduction of Vector-Valued Meromorphic Function
In 1980s, Ziegler 1 established Nevanlinna’s theory for the vector-valued meromorphic
function in finite dimensional spaces. After Ziegler some works related to vector-valued
meromorphic function were done in 1990s 2–4. In this section, we shall introduce the
following fundamental notations and results of vector-valued Nevanlinna theory which were
quoted from Ziegler 1.
We denote by Cn the usual n dimensional complex Euclidean space with the
coordinates w w1 , w2 , . . . , wn , the Hermitian scalar product
v, w v1 w1 v2 w2 · · · vn wn ,
v, w ∈ Cn ,
1.1
and the distance
v − w v − w, v − w1/2 .
1.2
2
Abstract and Applied Analysis
Let
w1 f1 z,
w2 f2 z, . . . , wn fn z
1.3
be n ≥ 1 complex valued functions of the complex variable z, which are meromorphic and
not all constant in the Gaussian plane C1 C, or in a finite disc
CR {|z| < R} ⊂ C,
0 < R < ∞.
1.4
Thus in CR , 0 < R ≤ ∞ we put C∞ C, a vector-valued meromorphic function
fz f1 z, f2 z, . . . , fn z
1.5
is given, which does not reduce to the constant zero vector 0 0, 0, . . . , 0. The jth derivative
j 1, 2, . . . of fz are defined by
j
j
j
f j z f1 z, f2 z, . . . , fn z .
1.6
For such a function, the notations “zero,” “pole,” and “multiplicity” are defined as
in the scalar case n 1 of only one meromorphic function f1 z. More explicitly, in the
punctured vicinity of each point z0 ∈ CR , the vector function w fz can developed into a
Laurent series
fz ck0 z − z0 k0 ck0 1 z − z0 k0 1 · · · ,
1.7
where the coefficients are vectors
ck ck1 , ck2 , . . . , ckn ∈ Cn ,
c k0 / 0, 0, . . . , 0.
1.8
In order to introduce the Nevanlinna theory of vector-valued meromorphic function,
we will denote by ”∞” the ideal element of the Aleksandrov one-point compactification of
Cn the two real infinities will be denoted by ∞ and −∞, resp.. Now, if k0 ≤ 0 in the above
Laurent expansion, then z0 will be called a pole or an ∞-point of fz of multiplicity −k0 ; in
such a point z0 at least one of the meromorphic component functions fj z has a pole of this
multiplicity in the ordinary sense of function theory, so that in z0 itself fz is not defined. If
k0 > 0 in Laurent expansion, then z0 is called a zero of fz of multiplicity k0 ; in such a point
z0 , all component functions fj z vanish, each with at least this multiplicity.
Abstract and Applied Analysis
3
Let nr, f or nr, ∞ denote the number of poles of fz in |z| ≤ r and nr, a denote
the number of a-points of fz in |z| ≤ r, counting with multiplicities. Define the volume
function associated with vector-valued meromorphic function fz,
r 1
log Δ logfξdx ∧ dy, ξ x iy
V r, ∞ V r, f 2π Cr
ξ
r 1
1
V r, a V r,
log Δ logfξ − adx ∧ dy, ξ x iy
f −a
2π Cr
ξ
1.9
and the counting function of finite or infinite a-points by
r n t, f − n 0, f
dt,
t
0
r
nt, ∞ − n0, ∞
Nr, ∞ n0, ∞ log r dt,
t
0
r
nt, a − n0, a
Nr, a n0, a log r dt,
t
0
N r, f n 0, f log r 1.10
respectively. Next, we define
1 2π
mr, ∞ m r, f log f reiθ dθ,
2π 0
1
1 2π
dθ,
mr, a log iθ f re − a
2π 0
1.11
T r, f m r, f N r, f .
Let nr, f or nr, ∞ denote the number of poles of fz in |z| ≤ r and nr, a denote the
number of a-points of fz in |z| ≤ r, ignoring multiplicities. Define the counting function of
finite or infinite a-points by
r n t, f − n 0, f
dt,
t
0
r
nt, ∞ − n0, ∞
dt,
Nr, ∞ n0, ∞ log r t
0
r
nt, a − n0, a
dt,
Nr, a n0, a log r t
0
N r, f n 0, f log r respectively.
1.12
4
Abstract and Applied Analysis
If fz is a vector-valued meromorphic function in the whole complex plane, then the
order and the lower order of fz are defined by
log T r, f
,
λ f lim sup
log r
r →∞
log T r, f
μ f lim inf
.
r →∞
log r
1.13
We call the vector-valued meromorphic function f admissible if
lim sup
r → ∞
T r, f
∞.
log r
1.14
Definition 1.1. For a meromorphic function fz vector-valued or scalar-valued, we denote
by Sr, f any quantity such that
S r, f o T r, f ,
r −→ ∞
1.15
without restriction if fz is of finite order and otherwise except possibly for a set of values
of r of finite linear measure.
Definition 1.1 quoted from 2. In 1, Ziegler established the following first main
theorem, logarithmic derivative lemma, and deficient values theorem for meromorphic
vector function.
Theorem A. Let fz f1 z, f2 z, . . . , fn z be a meromorphic vector function in CR . Then for
≡ a, then
0 < r < R ≤ ∞, a ∈ Cn , fz /
T r, f V r, a Nr, a mr, a O1.
1.16
Theorem B. Let fz f1 z, f2 z, . . . , fn z be a nonconstant meromorphic vector function in
C. Then
1
2π
2π
0
iθ f re dθ S r, f ,
log iθ f re − a
a ∈ Cn .
1.17
By the second main theorem, Ziegler 1 studies the following deficiency theorem for
meromorphic vector function. For any vector a ∈ Cn , we define the number δa δa, f by
putting
mr, a
V r, a Nr, a
,
δa δ a, f lim inf 1 − lim sup
r → ∞ T r, f
T r, f
r → ∞
m r, f
N r, f
δ∞ δ ∞, f lim inf 1 − lim sup ,
r → ∞ T r, f
r → ∞ T r, f
1.18
Abstract and Applied Analysis
5
and Θa Θa, f by putting
V r, a Nr, a
,
Θa Θ a, f 1 − lim sup
T r, f
r → ∞
N r, f
Θ∞ Θ ∞, f 1 − lim sup ,
r → ∞ T r, f
1.19
Theorem C. Let fz f1 z, f2 z, . . . , fn z be an admissible meromorphic vector function in
C. Then the set {a ∈ Cn ∪ {∞}, δa > 0} is at most countable and summing over all such points we
have
δa ≤
a
Θa ≤ 2.
1.20
a
2. A Fundamental Inequality of Meromorphic Vector Function
For meromorphic scalar-valued function fz, Milloux 5 has proved the following theorem.
Theorem D. If fz is a nonconstant meromorphic scalar-valued function in Gaussian complex plane
C and if ai , i 1, 2, . . . , q, are distinct elements of C (where q is any positive integer), then
q
qT r, f ≤ T r, f Nr, ai S r, f .
2.1
i1
For some alternative proofs of Theorem D, see 6 or 7. It is natural to consider
whether there exists a similar results if meromorphic scalar-valued function fz is replaced
by meromorphic vector-valued function fz. In this section, the main contribution is to
extend Theorem D to vector-valued meromorphic function by referring the method of 1, 7.
Theorem 2.1. Let fz f1 z, f2 z, . . . , fn z be an admissible meromorphic vector function in
C and if aj , j 1, 2, . . . , q, are distinct elements of Cn (where q is any positive integer), then
q qT r, f ≤ T r, f N r, aj V r, aj S r, f .
2.2
j1
Proof. Put
Fz q
1
.
fz − aj j1
2.3
We can get
1
2π
2π
log F re
0
iθ
dθ ≤ m r, 0, f
1
2π
2π
0
log F reiθ f reiθ dθ.
2.4
6
Abstract and Applied Analysis
Put
δ minai − aj .
i/
j
2.5
Let for the moment μ ∈ {1, 2, . . . , q} be fixed. Then we get in every point where
δ
δ
≤ ,
fz − aμ <
2q 4
2.6
3δ
,
fz − aν ≥ aμ − aν − fz − aμ ≥
4
2.7
the inequality
for μ / ν. Therefore, the set of points on ∂Cr which is determined by 2.6 is either empty or
any two such sets for different μ have empty intersection. In any case,
1
2π
2π
0
q 1 log F reiθ dθ ≥
log F reiθ dθ
2π μ1 fz−aμ <δ/2q, |z|r
q 1
1 dθ.
≥
log iθ f re − aμ 2π μ1 fz−aμ <δ/2q, |z|r
2.8
Because of
1
1
dθ
log iθ 2π fz−aμ <δ/2q, |z|r
f re − aμ 1
1
μ
dθ
m r, a
−
log iθ f re − aμ 2π fz−aμ ≥δ/2q, |z|r
2q
≥ m r, aμ − log ,
δ
2.9
it follows that
1
2π
2π
0
q
2q
log F reiθ dθ ≥
m r, aμ − log ,
δ
μ1
2.10
so that by 2.4
q
2q
1 2π
μ
≤ m r, 0, f m r, a
log F reiθ f reiθ dθ log .
2π
δ
0
μ1
2.11
Abstract and Applied Analysis
7
Thus by Theorem B, we have
q
m r, aμ ≤ m r, 0, f S r, f .
2.12
μ1
It follows from Theorem A that
m r, 0, f N r, 0, f V r, 0, f T r, f O1.
2.13
Thus from 2.12 and 2.13, we deduce
q
m r, aμ ≤ T r, f − N r, 0, f S r, f .
2.14
μ1
Adding
q
μ
μ1 Nr, a q
T
μ1
to both sides,
1
r,
f − aμ
q
≤ T r, f N r, aμ − N r, 0, f S r, f
μ1
q
T r, f N r, aμ − N0 r, 0, f S r, f ,
2.15
μ1
where N0 r, 0, f is formed with the zeros of f which are not zeros of any of f − aμ , i 1, 2, . . . , q. Since N0 r, 0, f ≥ 0, we have
q
1
r,
f − aμ
T
μ1
q
≤ T r, f N r, aμ S r, f .
2.16
μ1
Since
T
1
r,
f − aμ
V r, aμ T r, f O1,
2.17
it follows that
q qT r, f ≤ T r, f N r, aj V r, aj S r, f .
j1
2.18
8
Abstract and Applied Analysis
3. Characteristic Function of Derivative of
Meromorphic Vector Function
Let fz be a meromorphic scalar-valued function in C. The characteristic function of
derivative of fz with a δa 2 has been studied by Edrei 8, Shan and Singh 9, Singh
and Gopalakrishna 7, Singh and Kulkarni 10 and Weitsman 11. For example, Edrei 8
and Weitsman 11 have proved the following theorem.
Theorem E. Let fz be a transcendental meromorphic scalar-valued function of finite order and
assume a∈C δa η ≥ 1 and δ∞ 2 − η. Then
T r, f ∼ ηT r, f ,
r −→ ∞.
3.1
If a δa 2 is replaced by a Θa 2, Singh and Gopalakrishna 7 and Singh and
Kulkarni 10 have proved the following theorem.
Theorem F. Let fz be a transcendental meromorphic scalar-valued function of finite order and
assume a Θa 2. Then
T r, f 2 − Θ∞,
r → ∞ T r, f
lim
3.2
Nr, a
lim 1 − Θa
r → ∞ T r, f
for every a ∈ C ∪ {∞}.
It is natural to consider whether there exists a similar results if meromorphic scalarvalued function fz is replaced by meromorphic vector-valued function fz. In this section,
the main purpose is to extend the above theorems to vector-valued meromorphic function by
referring the method of 1, 7.
Theorem 3.1. Let fz f1 z, f2 z, . . . , fn z be an admissible meromorphic vector function of
finite order in C and assume a Θa 2. Then
T r, f lim 2 − Θ∞,
r → ∞ T r, f
N r, f
lim 1 − Θ∞,
r → ∞ T r, f
for every a ∈ Cn .
Nr, a V r, a
lim
1 − Θa
r → ∞
T r, f
3.3
Abstract and Applied Analysis
9
Proof. Now, basic estimates in vector-valued Nevanlinna theory 1 or 4 yields
T r, f m r, f N r, f ff m r,
N r, f f
f
m r, f N r, f N r, f
≤ m r,
f
f
.
≤ T r, f N r, f m r,
f
3.4
By Theorem B and the above inequality, we have
T r, f lim sup ≤ 2 − Θ∞.
r → ∞ T r, f
3.5
Let {aj } be a sequence of distinct vector in Cn containing all the vector of δaj > 0.
From Theorem 2.1, for any positive integer q, we have
q qT r, f ≤ T r, f N r, aj V r, aj S r, f .
3.6
j1
Hence
q
T r, f N r, aj V r, aj
S r, f
q ≤ lim inf lim sup
lim sup r → ∞ T r, f
T r, f
r → ∞ T r, f
j1 r → ∞
q T r, f S r, f
j
lim inf 1−Θ a
lim sup .
r → ∞ T r, f
r → ∞ T r, f
j1
3.7
Thus
q
T r, f Θ aj .
lim inf ≥
r → ∞ T r, f
j1
3.8
T r, f 2 − Θ∞ Θa ≤ lim inf .
r → ∞ T r, f
a∈Cn
3.9
Since q was arbitrary, we have
10
Abstract and Applied Analysis
This and 3.5 yield
T r, f 2 − Θ∞.
r → ∞ T r, f
lim
3.10
n
Let a ∈ Cn ∪ {∞} and {ai }∞
i1 an infinite sequence of distinct elements of C ∪ {∞}
n
a and Θb > 0. Then
which includes every b ∈ C ∪ {∞} satisfying b /
∞ Θ ai i1
Θb 2 − Θa.
b∈Cn ∪{∞}, b /
a
3.11
Let q be any integer ≥ 3. From Generalized Second Main Theorem see 1, Page 126,
we have
q−1 N r, ai V r, ai N r, f S r, f .
q − 2 T r, f 3.12
i1
Hence
q−1 N r, f
i
1−Θ a
lim inf .
q−2≤
r → ∞ T r, f
i1
3.13
q−1 N r, f
Θ ai − 1 ≤ lim inf .
r → ∞ T r, f
i1
3.14
Thus
Since this holds for all q ≥ 3, letting q → ∞ and combining 3.11, we get
1 − Θ∞ ∞ N r, f
N r, f
Θ ai − 1 ≤ lim inf ≤ lim sup 1 − Θ∞.
r → ∞ T r, f
r → ∞ T r, f
i1
3.15
So
N r, f
lim 1 − Θ∞.
r → ∞ T r, f
3.16
For every a ∈ Cn , Let q be any integer ≥ 3. From Generalized Second Main Theorem
see 1, Page 126, we have
q−2 N r, ai V r, ai Nr, a V r, a N r, f S r, f .
q − 2 T r, f i1
3.17
Abstract and Applied Analysis
11
Hence
q−2≤
q−2 i1
Nr, a V r, a
1 − Θ ai 1 − Θ∞ lim inf
.
r → ∞
T r, f
3.18
Thus
q−2 Nr, a V r, a
Θ ai Θ∞ − 1 ≤ lim inf
.
r → ∞
T r, f
i1
3.19
Since this holds for all q ≥ 3, letting q → ∞ and combining 3.11, we get
1 − Θa ∞ Nr, a V r, a
Θ ai − 1 ≤ lim inf
r → ∞
T r, f
i1
Nr, a V r, a
1 − Θa.
≤ lim sup
T r, f
r → ∞
3.20
So
lim
r → ∞
Nr, a V r, a
1 − Θa.
T r, f
3.21
From Theorem 3.1, we have the following corollary
Corollary 3.2. Let fz f1 z, f2 z, . . . , fn z be an admissible meromorphic vector function of
finite order in C and assume a∈Cn Θa 2. Then
T r, f ∼ 2T r, f ,
r −→ ∞.
3.22
Corollary 3.3. Let fz f1 z, f2 z, . . . , fn z be an admissible meromorphic vector function of
finite order in C and assume a∈Cn Θa η ≥ 1 and δ∞ 2 − η. Then
T r, f ∼ ηT r, f ,
r −→ ∞.
3.23
Corollary 3.4. Let fz f1 z, f2 z, . . . , fn z be an admissible meromorphic vector function of
finite order in C and assume a δa 2. Then
T r, f lim 2 − δ∞,
r → ∞ T r, f
N r, f
lim 1 − δ∞,
r → ∞ T r, f
for every a ∈ Cn .
Nr, a V r, a
lim
1 − δa
r → ∞
T r, f
3.24
12
Abstract and Applied Analysis
Proof. Since δa ≤ Θa for every a ∈ Cn ∪{∞} and Theorem C, it follows that, if
then a Θa 2 and δa Θa for every a ∈ Cn ∪ {∞}. Hence
T r, f lim 2 − δ∞
r → ∞ T r, f
a δa
2,
3.25
follows by Theorem 3.1.
Now, for every a ∈ Cn ,
lim
r → ∞
Nr, a V r, a
1 − Θa 1 − δa.
T r, f
3.26
Further
Nr, a ≤ Nr, a.
3.27
Hence
1 − δa ≤ lim
r → ∞
Nr, a V r, a
T r, f
≤ lim inf
r → ∞
Nr, a V r, a
T r, f
≤ lim sup
r → ∞
3.28
Nr, a V r, a
T r, f
1 − δa.
Similarly,
N r, f
lim 1 − Θ∞ 1 − δ∞.
r → ∞ T r, f
3.29
Nr, ∞ ≤ Nr, ∞.
3.30
Further
Hence
1 − δ∞ ≤ lim
r → ∞
Nr, ∞
Nr, ∞
Nr, ∞
≤ lim inf ≤ lim sup 1 − δ∞.
r
→
∞
T r, f
T r, f
r → ∞ T r, f
From Corollary 3.4, we have the following corollary.
3.31
Abstract and Applied Analysis
13
Corollary 3.5. Let fz f1 z, f2 z, . . . , fn z be an admissible meromorphic vector function of
finite order in C and assume a∈Cn δa 2. Then
T r, f ∼ 2T r, f ,
r −→ ∞.
3.32
Corollary 3.6. Let fz f1 z, f2 z, . . . , fn z be an admissible meromorphic vector function of
finite order in C and assume a∈Cn δa η ≥ 1 and δ∞ 2 − η. Then
T r, f ∼ ηT r, f ,
r −→ ∞.
3.33
Acknowledgments
This research was partially supported by the NSF of Jiangxi Province Grant 2010GZC0187,
by NSF of Educational Department of the Hubei Province Grant T201009, Q20112807, and
by Grant PY1002 of Xianning University.
References
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