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Abstract and Applied Analysis
Volume 2012, Article ID 906893, 8 pages
doi:10.1155/2012/906893
Research Article
Uniqueness Theorems on Entire Functions and
Their Difference Operators or Shifts
Baoqin Chen,1 Zongxuan Chen,1 and Sheng Li2
1
2
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
College of Science, Guangdong Ocean University, Zhangjiang 524088, China
Correspondence should be addressed to Zongxuan Chen, chzx@vip.sina.com
Received 23 December 2011; Accepted 24 March 2012
Academic Editor: István Györi
Copyright q 2012 Baoqin Chen et al. 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.
We study the uniqueness problems on entire functions and their difference operators or shifts. Our
main result is a difference analogue of a result of Jank-Mues-Volkmann, which is concerned with
the uniqueness of the entire function sharing one finite value with its derivatives. Two relative
results are proved, and examples are provided for our results.
1. Introduction and Main Results
Throughout this paper, we assume the reader is familiar with the standard notations and
fundamental results of Nevanlinna theory of meromorphic functions see, e.g., 1–3. In
what follows, a meromorphic function always means meromorphic in the whole complex
plane, and c always means a nonzero complex constant. For a meromorphic function fz,
we define its shift by fz c and its difference operators by
Δc fz fz c − fz,
Δnc fz Δn−1
Δc fz ,
c
n ∈ N, n ≥ 2.
1.1
For a meromorphic function fz, we use Sf to denote the family of all meromorphic
functions az that satisfy T r, a Sr, f, where Sr, f oT r, f, as r → ∞ outside of
a possible exceptional set of finite logarithmic measure. Functions in the set Sf are called
small functions with respect to fz.
Let fz and gz be two meromorphic functions, and let az be a small function with
respect to fz and gz. We say that fz and gz share az IM, provided that fz − az
and gz − az have the same zeros ignoring multiplicities, and we say that fz and gz
share az CM, provided that fz − az and gz − az have the same zeros with the same
multiplicities.
2
Abstract and Applied Analysis
Uniqueness theory of meromorphic functions is an important part of the Nevanlinna
theory. In the past 40 years, a very active subject is the investigation on the uniqueness of the
entire function sharing values with its derivatives, which was initiated by Rubel and Yang
4. We first recall the following result by Jank et al. 5.
Theorem A see 5. Let f be a nonconstant meromorphic function, and let a ≡
/ 0 be a finite
constant. If f, f , and f share the value a CM, then f ≡ f .
Recently, value distribution in difference analogues of meromorphic functions has
become a subject of some interest see, e.g., 6–11. In particular, a few authors started to
consider the uniqueness of meromorphic functions sharing small functions with their shifts
or difference operators see, e.g., 12, 13.
In this paper, we consider difference analogues of Theorem A.
Theorem 1.1. Let fz be a nonconstant entire function of finite order, and let az /
≡ 0 ∈ Sf be
a periodic entire function with period c. If fz, Δc f, and Δ2c f share az CM, then Δ2c f Δc f.
Example 1.2. Let fz ez ln 2 and c 1. Then, for any a ∈ C, we notice that fz, Δc f, and
Δ2c f share a CM and can easily see that Δ2c f Δc f. This example satisfies Theorem 1.1.
Remark 1.3. In Example 1.2, we have Δ2c f Δc f f. However, it remains open whether
the claim Δ2c f Δc f in Theorem 1.1 can be replaced by Δc f f in general. In fact, the
next example resulted from our efforts to find an entire function fz satisfying Theorem 1.1,
f.
while Δc f /
Example 1.4. Let fz ez ln 2 − 2, a −1, b 1, and c 1. Then we observe that fz − a ez ln 2 − 1, Δc f − b ez ln 2 − 1, and Δ2c f − b ez ln 2 − 1 share 0 CM. Here, we also get Δ2c f Δc f.
From this example, it is natural to ask what happens if fz − az, Δc f − bz, and
Δ2c f − bz share 0 CM, where az and bz are two not necessarily distinct small periodic
entire functions. Considering this question, we prove the following Theorem 1.5, whose proof
is omitted as it is similar to the proof of Theorem 1.1.
Theorem 1.5. Let fz be a nonconstant entire function of finite order, and let az, bz /
≡ 0 ∈
Sf be periodic entire functions with period c. If fz − az, Δc f − bz, and Δ2c f − bz share 0
CM, then Δ2c f Δc f.
Now it would be interesting to know what happens if the difference operators of fz
are replaced by shifts of fz in Theorem 1.5. We prove the following result concerning this
question.
Theorem 1.6. Let fz be a nonconstant entire function of finite order, let az, bz ∈ Sf be two
distinct periodic entire functions with period c, and let n and m be positive integers satisfying n > m.
If fz − az, fz mc − bz, and fz nc − bz share 0 CM, then fz mc fz nc for
all z ∈ C.
Example 1.7. Let fz sin z 1, a 0, b 2, and c π. Then we notice that fz − a sin z 1, fz c − b − sin z − 1, and fz 3c − b − sin z − 1 share 0 CM and can easily
see that fz c fz 3c for all z ∈ C. This example satisfies Theorem 1.6.
Abstract and Applied Analysis
3
Example 1.8. Let fz ez az, where az ∈ Sf is a periodic entire function with period
2
2
2
1. Then fz − az ez , fz 1 − az ez1 , and fz 3 − az ez3 share 0 CM,
while fz c − fz 3c /
≡ 0. This example shows that the condition that az and bz are
distinct in Theorem 1.6 cannot be deleted.
2
2. Proof of Theorem 1.1
Lemma 2.1 see 8, Theorem 2.1. Let fz be a meromorphic function of finite order ρ and let c
be a nonzero complex constant. Then, for each ε > 0,
T r, fz c T r, fz O r ρ−1ε O log r .
2.1
Lemma 2.2 see 10, Lemma 2.3. Let c ∈ C, n ∈ N, and let fz be a meromorphic function of
finite order. Then for any small periodic function az with period c, with respect to fz,
Δn f
m r, c
S r, f ,
f −a
2.2
where the exceptional set associated with Sr, f is of at most finite logarithmic measure.
Proof of Theorem 1.1. Suppose, on the contrary, the assertion that Δ2c f /
Δc f. Note that fz is
a nonconstant entire function of finite order. By Lemma 2.1, Δc f and Δ2c f are entire functions
of finite order.
Since fz, Δc f, and Δ2c f share az CM, then we have
Δ2c f − az
eαz ,
fz − az
Δc f − az
eβz ,
fz − az
2.3
where αz and βz are polynomials.
Set
ϕz Δ2c f − Δc f
.
fz − az
2.4
From 2.3, we get ϕz eαz − eβz . Then by supposition and 2.4, we see that
ϕz ≡
/ 0. By Lemma 2.2, we deduce that
T r, ϕ m r, ϕ
Δ2c f
≤ m r,
fz − az
m r,
Δc f
fz − az
log 2 S r, f .
2.5
4
Abstract and Applied Analysis
Note that eα /ϕ − eβ /ϕ 1. By using the second main theorem and 2.5, we have
α
α
α
ϕ
e
e
e
1
T r,
≤ N r,
N r, α N r, α
S r,
ϕ
ϕ
e
e /ϕ − 1
ϕ
α
α
ϕ
ϕ
e
e
N r, α N r, β S r,
N r,
ϕ
e
ϕ
e
α
e
.
S r, f S r,
ϕ
2.6
Thus, by 2.5 and 2.6, we have T r, eα Sr, f. Similarly, T r, eβ Sr, f.
By Lemma 2.2 and the first equation in 2.3, we deduce that az/fz − az Δ2c f/fz − az − eαz and
1
m r,
fz − az
Δ2c f
1
αz
m r,
−e
az fz − az
Δ2c f
m r, eαz S r, f
≤ m r,
fz − az
S r, f .
2.7
From 2.7, we see that
N r,
1
fz − az
T r, fz − m r,
1
fz − az
T r, fz S r, f .
S r, f
2.8
Now we rewrite the second equation in 2.3 as Δc f eβz fz − az az and
deduce that
Δ2c f Δc eβz fz − az az
eβzc fz c − az c az c − eβz fz − az − az
eβzc fz c − az − eβz fz − az .
2.9
This together with the first equation in 2.3 gives
fz c eαz−βzc eβz−βzc fz
− az eαz−βzc eβz−βzc − 1 − e−βzc ,
2.10
Abstract and Applied Analysis
5
that is,
Δc f eαz−βzc eβz−βzc − 1 fz
2.11
− az eαz−βzc eβz−βzc − 1 − e−βzc .
Thus, 2.11 can be rewritten as
Δc f γzfz δz,
2.12
γz eαz−βzc eβz−βzc − 1,
δz −az eαz−βzc eβz−βzc − 1 − e−βzc
2.13
where
−azγz aze−βzc ,
which satisfy T r, γ Sr, f and T r, δ Sr, f.
Now we rewrite Δc f γzfz δz as
Δc f − az − γz fz − az γzaz δz − az.
2.14
Suppose that γzaz δz − az /
≡ 0. Let z0 be a zero of fz − az with multiplicity
k. Since fz, Δc f share az CM, then z0 is a zero of Δc f − az with multiplicity k. Thus,
z0 is a zero of Δc f − az − γzfz − az with multiplicity at least k. Then, by 2.8 and
2.14, we see that
1
N r,
γzaz δz − az
1
N r,
Δc f − az − γz fz − az
1
≥ N r,
fz − az
T r, fz S r, f .
2.15
On the other hand, we have
N r,
1
γzaz δz − az
≤ T r,
1
γzaz δz − az
S r, f .
2.16
Then, by 2.15 and 2.16, we get T r, f ≤ Sr, f, which is a contradiction.
Thus, γzazδz−az ≡ 0. Noting that δz −azγzaze−βzc , we deduce
−βzc
≡ 1. So, eβz ≡ eβzc ≡ 1, since βz is a polynomial.
that e
By the second equation in 2.3, we obtain Δc f f, which leads to Δ2c f Δc f. This is
a contradiction. The proof is thus completed.
6
Abstract and Applied Analysis
3. Proof of Theorem 1.6
Lemma 3.1 see 9, Corollary 2.2. Let fz be a nonconstant meromorphic function of finite order,
c ∈ C and δ < 1. Then
fz c
m r,
fz
T r |c|, f
,
o
rδ
3.1
for all r outside of a possible exceptional set with finite logarithmic measure.
Proof of Theorem 1.6. Suppose, on the contrary, the assertion that fzmc−fznc /
≡ 0. Since
fz − az, fz mc − bz, and fz nc − bz share 0 CM, then we have
fz nc − bz
eαz ,
fz − az
fz mc − bz
eβz ,
fz − az
3.2
where αz and βz are polynomials.
By 3.2, we obtain
fz nc − fz mc
eαz − eβz .
fz − az
3.3
Set ψz eαz − eβz . Then by supposition, we see that ψz /
≡ 0. By Lemma 3.1, we
deduce that
fz nc − fz mc
T r, ψ m r,
fz − az
fz nc − az nc
fz mc − az mc
≤ m r,
m r,
log 2
fz − az
fz − az
S r, f .
3.4
Note that eα /ψ −eβ /ψ 1. Thus, using a similar method as in the proof of Theorem 1.1,
we get T r, eα Sr, f and T r, eβ Sr, f.
By Lemma 3.1 and the first equation in 3.2, we deduce that
m r,
1
fz − az
m r,
fz nc − az
1
− eαz
bz − az
fz − az
fz nc − az nc
mr, eα S r, f
≤ m r,
fz − az
S r, f .
3.5
Abstract and Applied Analysis
7
From 3.5, we see that
N r,
1
fz − az
T r, fz S r, f .
3.6
Now we rewrite the second equation in 3.2 as fz mc eβz fz − az bz
and deduce that
fz nc eβzn−mc fz n − mc − az n − mc
bz n − mc.
3.7
This together with the first equation in 3.2 gives
fz n − mc eαz−βzn−mc fz
− azeαz−βzn−mc az n − mc,
3.8
that is,
fz nc eαzmc−βznc fz mc
− az mceαzmc−βznc az nc
3.9
eαzmc−βznc fz mc − azeαzmc−βznc az.
Now we rewrite 3.9 as
fz nc − bz − eαzmc−βznc fz mc − bz
bz − az eαzmc−βznc − 1 .
3.10
Suppose that eαzmc−βznc ≡ 1; then, by 3.9, we get fz nc fz mc, which is
a contradiction.
≡ 0. Then using a similar method as in the proof of
Now we have eαzmc−βznc − 1 /
Theorem 1.1, we can also get a contradiction and obtain that fz mc fz nc for all
z ∈ C. Thus, Theorem 1.6 is proved.
Acknowledgments
This work was supported by the National Natural Science Foundation of China no.
11171119. The authors would like to thank the referee for valuable suggestions.
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Abstract and Applied Analysis
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