Volume 2009, Article ID 208516,10pages doi:10.1155/2009/208516
Research Article
On Uniqueness of Meromorphic Functions with
Multiple Values in Some Angular Domains
Zu-Xing Xuan
1, 21Department of Mathematical Sciences, Tsinghua University, Haidian District, Beijing 100084, China 2Department of Basic Courses, Beijing Union University, No. 97 Bei Si Huan Dong Road,
Chaoyang District, Beijing 100101, China
Correspondence should be addressed to Zu-Xing Xuan,[email protected]
Received 4 March 2009; Accepted 30 June 2009
Recommended by Narendra Kumar Govil
This article deals with problems of the uniqueness of transcendental meromorphic function with shared values in some angular domains dealing with the multiple values which improve a result of J. Zheng.
Copyrightq2009 Zu-Xing Xuan. 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.
1. Introduction
A transcendental meromorphic function is meromorphic in the complex plane C and not rational. We assume that the readers are familiar with the Nevanlinna theory of meromorphic functions and the standard notations such as Nevanlinna deficiency δa, f of fz with respect toa ∈ C and Nevanlinna characteristicTr, foffz. And the lower orderμand the orderλare in turn defined as follows:
μμflim inf
r→ ∞
logTr, f
logr ,
λλflim sup
r→ ∞
logTr, f
logr .
1.1
if they have five distinct IM shared values in X C. After his very fundamental work, the uniqueness of meromorphic functions with shared values in the whole complex plane attracted many investigationssee3. Recently, Zheng in4suggested for the first time the investigation of uniqueness of a function meromorphic in a precise subset ofC, and this is an interesting topic.
Givenmpair of real numbers{αj, βj}satisfying
−π ≤α1< β1≤α2< β2≤ · · · ≤αm< βm≤π, 1.2
we define
ωmax
π β1−α1, . . . ,
π βm−αm
. 1.3
Zheng in4proved the following theorem.
Theorem A. Letfzandgzbe both transcendental meromorphic functions, and letfzbe of finite orderλand such that for somea∈C and an integerp ≥0, δ δa, fp>0. Formpair of
real numbers{αj, βj}satisfying1.2and
m
j1
αj1−βj
< 4
σarcsin
δ
2, 1.4
whereσmax{ω, μ}, assume thatfzandgzhave five distinct IM shared values inX mj1{z:
αj≤argz≤βj}. Ifω < λf, thenfz≡gz.
However, it was not discussed whether there are similar results dealing with multiple values in some angular domains. In this paper we investigate this problem.
We useEka, X, fto denote the set of zeros offz−ainX, with multiplicities no
greater thank, in which each zero counted only once. Our main result is what follows.
Theorem 1.1. Letfzandgzbe both transcendental meromorphic functions, and letfzbe of finite orderλand such that for somea∈C and an integerp ≥0, δ δa, fp>0. Formpair of
real numbers{αj, βj}satisfying1.2and
m
j1
αj1−βj
< 4
σarcsin
δ
where σ max{ω, μ}, assume that aj j 1,2, . . . , q areq distinct complex numbers, and let
kj j 1,2, . . . , qbe positive integers or∞satisfying
k1≥k2 ≥ · · · ≥kq, 1.6
Ekjaj, X, f
Ekjaj, X, g
, 1.7
q
j3
kj
kj1
>2, 1.8
whereX qj1{z:αj ≤argz≤βj}. Ifω < λf,thenfz≡gz.
2. Proof of
Theorem 1.1
First we introduce several lemmas which are crucial in our proofs. The following result was proved in5 also see6.
Lemma 2.1see5. Letfzbe transcendental and meromorphic inCwith the lower order0 ≤
μ <∞and the order0< λ≤ ∞. Then for arbitrary positive numberσsatisfyingμ≤σ≤λand a set
Ewith finite linear measure, there exists a sequence of positive numbers{rn}such that
1rn∈E,limn→ ∞rn/n ∞,
2lim infn→ ∞logTrn, f/logrn≥σ,
3Tt, f<1o1t/rn σTrn, f, t∈rn/n, nrn.
A sequencernsatisfying1,2, and3inLemma 2.1is called Polya peak of orderσ
outsideEin this article. Forr >0 anda∈Cdefine
Dr, a:
θ∈−π, π: log 1
freiθ−a >
1 logrT
r, f
, 2.1
Dr,∞:
θ∈−π, π: logfreiθ> 1
logrT
r, f. 2.2
The following result is a special version of the main result of Baernstein7.
Lemma 2.2. Letfzbe transcendental and meromorphic inCwith the finite lower orderμand the order0 < λ ≤ ∞and for somea ∈ C, δ δa, f > 0. Then for arbitrary Polya peakrn of order
σ >0, μ≤σ≤λ, we have
lim inf
n→ ∞ mes Drn, a≥min ⎧ ⎨ ⎩2π,
4
σarcsin
δ
2
⎫ ⎬
⎭. 2.3
Nevanlinna theory on angular domain will play a key role in the proof of theorems. Letfzbe a meromorphic function on the angular domainΩα, β {z : α ≤ argz ≤ β}, where 0< β−α≤2π. Nevanlinna defined the following notationssee8:
Aα,β
r, f ω π
r
1
1
tω −
tω
r2ω
logfteiαlogfteiβdt t ,
Bα,β
r, f 2ω πrω
β
α
logfreiθsinωθ−αdθ,
Cα,β 2
1<|bn|<r
1
|bn|ω −
|bn|ω
r2ω
sinωθn−α,
2.4
whereω π/β−αandbn |bn|eiθn are the poles offzonΩα, βappearing according
to their multiplicities.Cα,βr, fis called the angular counting function of the poles off on
Ωα, βand Nevanlinna’s angular characteristic is defined as follows:
Sα,β
r, fAα,β
r, fBα,β
r, fCα,β
r, f. 2.5
Throughout, we denote byRα,βr,∗a quantity satisfying
Rα,βr,∗ O
logrSα,βr,∗
, r∈E, 2.6
whereEdenotes a set of positive real numbers with finite linear measure. It is not necessarily the same for every occurrence in the context9.
Lemma 2.3. Letfzbe meromorphic onΩα, β. Then for arbitrary complex numbera, we have
Sα,β
1
f−a
Sα,β
r, fO1, 2.7
and for an integerp≥0,
Sα,β
r, fp≤2pSα,β
r, fRα,β
r, f,
Aα,β
r,f
p
f
Bα,β
r,f
p
f
Rα,β
r, f,
2.8
Lemma 2.4. Letfzbe meromorphic onΩα, β. Then for arbitraryqdistinctaj ∈C 1≤j ≤q,
we have
q−2Sα,β
r, f≤
q
j1
Cα,β
r, 1
f−aj
Rr, f, 2.9
where the termCα,βr,1/f−ajwill be replaced byCα,βr, fwhen someaj∞.
We useCkα,βr,1/f−ato denote the zeros offz−ainΩα, βwhose multiplicities are no greater thankand are counted only once. Likewise, we useCα,βk1r,1/f−ato denote the zeros offz−ainΩα, βwhose multiplicities are greater thankand are counted only once.
Lemma 2.5. Letfzbe meromorphic onΩα, β,and letkjj 1,2, . . . , qbeqpositive integers.
Then for arbitraryqdistinctaj ∈C 1≤j ≤q, we have
i q−2Sα,β
r, f<
q
j1
kj
kj1
Ckα,βj
r, 1 f−aj
q
j1
1
kj1
Cα,β
r, 1
f−aj
Rr, f,
ii
⎛
⎝q−2−q
j1
1
kj1
⎞
⎠Sα,βr, f<q
j1
kj
kj1
Ckα,βj
r, 1 f−aj
Rr, f,
2.10
where the termCα,βr,1/f−ajwill be replaced byCα,βr, fwhen someaj∞.
Proof. According to our notations, we have
Cα,β
r, 1
f−a
Ckα,β
r, 1 f−a
Cα,βk1
r, 1
f−a
k k1C
k α,β
r, 1
f−a
1 k1C
k α,β
r, 1
f−a
Cα,βk1
r, 1 f−a
≤ k
k1C
k α,β
r, 1
f−a
1 k1C
k α,β
r, 1
f−a
1 k1C
k1
α,β
r, 1
f−a
k k1C
k α,β
r, 1
f−a
1 k1Cα,β
r, 1
f−a
.
ByLemma 2.4,
q−2Sα,β
r, f≤
q
j1
Cα,β
r, 1
f−aj
Rr, f
≤q
j1
kj
kj1C kj α,β
r, 1
f−aj
q
j1
1
kj1Cα,β
r, 1
f−aj
Rr, f,
2.12
andifollows.
Furthermore,Cα,βr,1/f−aj < Sα,βr, f, and on combining this withi, we get
ii.
Proof ofTheorem 1.1. Supposefz/≡gz. For convenience, below we omit the subscript of all the notations, such asSr,∗andCr,∗. By applyingLemma 2.5togand1.6, we have
⎛ ⎝q
j3
kj
kj1
2k2
k21 −
2
⎞
⎠Sr, g≤ k2 k21
q
j1
Ckj
r, 1 g−aj
Rr, g
≤ k2
k21
C
r, 1 f−g
Rr, g
≤ k2
k21S
r, f−gRr, g
≤ k2
k21S
r, f k2 k21S
r, gRr, g,
2.13
so that
⎛ ⎝q
j3
kj
kj1
k2
k21−2
⎞
⎠Sr, g−Rr, g< k2
k21S
r, f. 2.14
This implies thatRr, g Rr, f. We have also2.14for alternation offandg, then
⎛ ⎝q
j3
kj
kj1
k2
k21−
2
⎞
⎠Sr, f−Rr, f< k2
k21
Sr, g≤Sr, fRr, f. 2.15
By1.8, we have
We assume thata∈ C. By the same argument we can showTheorem 1.1for the case whena∞. By applyingLemma 2.3and2.16, we estimate
B
r, 1 fp−a
≤Sr, fpO1
AB
r,f
p
f
ABr, fpCr, fCr, fO1
≤p1Sr, fRr, fOlogr, r /∈E.
2.17
The following method comes from10. But we quote it in detail here because of its independent significance. Note thatλf> ω. We need to treat two cases.
Iλf > μ.Thenλfp λf> σ ≥ μ μfp. And by the inequality1.5, we
can take a real number >0 such that
m
j1
αj1−βj2
2 < 4
σ2arcsin
δ
2, 2.18
whereαm12πα1, and
λfp> σ2 > μ. 2.19
ApplyingLemma 2.1tofpzgives the existence of the Polya peakrnof orderσ2offp
such thatrn/∈E, and then fromLemma 2.2for sufficiently largenwe have
mesDrn, a> 4
σ2arcsin
δ
2 −, 2.20
sinceσ2 >1/2. We can assume for all then,13holds. Set
K:mes ⎛
⎝Drn, a"#m
j1
αj, βj−
⎞⎠
. 2.21
Then from2.18and2.20it follows that
K≥mesDrn, a−mes
⎛
⎝0,2π\#m
j1
αj, βj−
⎞⎠
mesDrn, a−mes
⎛ ⎝#m
j1
βj−, αj1
⎞⎠
mesDrn, a− m
j1
αj1−βj2
> >0.
It is easy to see that there exists aj0such that for infinitely manyn, we have
mesDrn, a"αj0, βj0−
> K
q. 2.23
We can assume for all then,2.23holds. SetEnDrn, a
$
α−j0, βj0−. Thus from the
definition2.1ofDr, ait follows that
βj 0−
αj0
log 1
fprneiθ−adθ≥
En
log 1
fprneiθ−adθ
≥mesEn
Trn, fp
logrn
> K m
Trn, fp
logrn
.
2.24
On the other hand, by the definition2.4ofBα,βr,∗and2.14, we have
βj 0−
αj0
log 1
fprneiθ−adθ≤
π
2ωj0sin
ωj0 rωj0Bα
j0,βj0
r, 1
fp−a
<K%j0r
ωj0logr, r /∈E.
2.25
Combining2.24with2.25gives
Trn, fp
≤ mK%j0 K r
ωj0
n log2rn. 2.26
Thus from1.5inLemma 2.1forσ2, we have
σ≤lim sup
n→ ∞
logTrn, fp
logrn ≤
ωj0 ≤σ. 2.27
This is impossible.
II λf μ.Thenσ μλf λfp μfp.By the same argument as inI
with all theσ2replaced byσμ, we can derive
maxω, μσ≤ω < λf. 2.28
This is impossible.Theorem 1.1follows.
Remark 2.6. In Theorem A,q5, k1k2 k3k4k5∞,then
k3
k31
k4
k41
k5
k51
so Theorem A is a special case ofTheorem 1.1. Meanwhile, Zheng in4, pages 153–154gave some examples to indicate that the conditions are necessary. So the conditions in theorem are also necessary.
Corollary 2.7. InTheorem 1.1,
iifq7, thenfz≡gz,
iiifq6, k3≥2,thenfz≡gz,
iiiifq5, k3≥3, k5≥2, thenfz≡gz,
ivifq5, k4≥4, thenfz≡gz,
vifq5, k3≥5, thenfz≡gz,
viifq5, k3≥6, k4≥2, thenfz≡gz,
Corollary 2.8. Letfzandgzbe both transcendental meromorphic functions and letfzbe of finite lower orderμand such that for somea∈C and an integerp ≥0, δ δa, fp >0. Form
pair of real numbers{αj, βj}satisfying1.2and
m
j1
αj1−βj
< 4
σarcsin
δ
2, 2.30
whereσmax{ω, μ}, assume thatajj1,2, . . . , qareq5 2/kdistinct complex numbers
satisfying thatEkaj, X, f Ekaj, X, g j 1,2, . . . , q, wherek is an integer or∞. If ω <
λf, thenfz≡gz.
Corollary 2.9. Letfzandgzbe both transcendental meromorphic functions and letfzbe of finite lower orderμand such that for somea∈C and an integerp ≥0, δ δa, fp >0. Form
pair of real numbers{αj, βj}satisfying1.2and
m
j1
αj1−βj
< 4
σarcsin
δ
2, 2.31
whereσmax{ω, μ}, assume thatajj1,2, . . . , qareq5distinct complex numbers satisfying
thatE3aj, X, f E3aj, X, g j1,2,3,E2aj, X, f E2aj, X, g j4,5, thenfz≡
gz.
Question 1. For two meromorphic functions defined in C, there are many uniqueness theorems when they share small functions az is called a small function of fz if
Tr, az oTr, fr → ∞ see3. So we ask an interesting question: are there similar results when they share small functions in some precise domainX⊆C?
Acknowlegment
References
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