R E S E A R C H
Open Access
On value distribution and uniqueness of
meromorphic function with finite logarithmic
order concerning its derivative and
q-shift
difference
Xiu-Min Zheng
1*and Hong-Yan Xu
2*Correspondence:
1Institute of Mathematics and
Information Science, Jiangxi Normal University, Nanchang, 330022, China Full list of author information is available at the end of the article
Abstract
In this paper, we study the value distribution of a meromorphic functionf(z) concerning its derivativef(z) andq-shift differencef(qz+c), wheref(z) is of finite logarithmic order. We also investigate the uniqueness of differential-q-shift-difference polynomials with more general forms of entire functions of order zero.
Keywords: differential-q-shift-difference; meromorphic function; logarithmic order
1 Introduction and main results
The fundamental theorems and the standard notations of the Nevanlinna value distribu-tion theory of meromorphic funcdistribu-tions will be used (seee.g.Hayman [], Yang [] and Yi and Yang []). In addition, for a meromorphic functionf(z), we useS(r,f) to denote any quantity satisfyingS(r,f) =o(T(r,f)) for allroutside a possible exceptional setEof finite logarithmic measurelimr→∞(,r]∩E
dt
t <∞and also useS(r,f) to denote any quantity
sat-isfyingS(r,f) =o(T(r,f)) for allron a setFof logarithmic density , where the logarithmic density of a setFis defined by
lim
r→∞
logr
(,r]∩F
tdt.
The order of a meromorphic functionf(z) is defined by
ρ(f) =lim sup
r→∞
logT(r,f) logr .
The logarithmic order of a meromorphic functionf(z) is defined by (see [])
ρlog(f) =lim sup
r→∞
logT(r,f) log logr .
Ifρlog(f) <∞, thenf(z) is said to be of finite logarithmic order. It is clear that if a mero-morphic functionf(z) has finite logarithmic order, thenf(z) has order zero.
Remark .[, ] Iff(z) is a meromorphic function of finite positive logarithmic order
ρlog(f), thenT(r,f) has proximate logarithmic orderρlog(f). The logarithmic-type function ofT(r,f) is defined asU(r,f) = (logr)ρlog(f). We haveT(r,f)≤U(r,f) for sufficiently larger. The logarithmic exponent of convergence ofa-points off(z) is equal to the logarithmic order ofn(r,f–a), which is defined as
λlog(a) =lim sup
r→∞
logn(r,
f–a)
log logr .
We see by [] that for a meromorphic functionf(z) of finite positive logarithmic order
ρlog(f), the logarithmic order ofN(r,f–a) isλlog(a) + , whereλlog(a) is the logarithmic order ofn(r,f–a).
Moreover, we assume in the whole paper thatm,n,k,tm,tnare positive integers,q∈
C\{},c∈C, anda(z) is a non-zero small function with respect tof(z), that is,a(z) is a non-zero meromorphic function of growthS(r,f).
Many mathematicians were interested in the value distribution of different expressions of meromorphic functions and obtained lots of important theorems (seee.g.[, –]). Es-pecially, Hayman [] discussed Picard’s values of meromorphic functions and their deriva-tives, and he obtained the following famous theorem in .
Theorem .[] Let f(z)be a transcendental entire function.Then
(a) forn≥,f(z)f(z)nassumes all finite values except possibly zero infinitely often; (b) forn≥anda= ,f(z) –af(z)nassumes all finite values infinitely often.
Further, for a transcendental meromorphic functionf(z), Chen and Fang [] obtained the following result.
Theorem .[, Theorem ] Let f(z)be a transcendental meromorphic function.If n≥ is a positive integer,f(z)f(z)n– has infinitely many zeros.
Recently, with the establishments of difference analogies of the Nevanlinna theory (see e.g. [–]), many mathematicians focused on studying difference analogies of Theo-rems . and .. The main purpose of these results (see e.g. [, –]) is to get the sharp estimation of the value ofnto make difference polynomialsf(z+c)f(z)n–aand f(z+c) –af(z)n–badmit infinitely many zeros.
Meantime,q-difference analogies of the Nevanlinna theory and their applications on the value distribution of q-difference polynomials andq-shift-difference equations are also studied (seee.g.[–]). Especially, for a transcendental meromorphic (resp. entire) functionf(z) of order zero, Zhang and Korhonen [] studied the value distribution ofq -difference polynomials off(z) and found that ifn≥ (resp.n≥), thenf(qz)f(z)nassumes
every non-zero valuea∈Cinfinitely often (see [, Theorem .]).
Further, Xu and Zhang [] investigated the zeros of q-shift difference polynomials of meromorphic functions of finite logarithmic order and obtained the following result in .
ρlog(f) – and q,c are non-zero complex constants,then for n≥,f(z)nf(qz+c)assumes every value b∈C infinitely often.
One main aim of this paper is to investigate the zeros of differential-q-shift-difference polynomials aboutf(z),f(z), andf(qz+c), wheref(z) is of finite positive logarithmic order.
Theorem . Let f(z)be a transcendental meromorphic function of finite logarithmic or-derρlog(f),with the logarithmic exponent of convergence of poles less thanρlog(f) – .Set
F(z) =f(qz+c)nf(z).
If n≥,then F(z) –a(z)has infinitely many zeros.
We also deal with the value distribution of a differential-q-shift-difference polynomial with another form aboutf(z),f(z) andf(qz+c), and we obtain the following result.
Theorem . Let f(z)be a transcendental meromorphic function of finite logarithmic or-derρlog(f),with the logarithmic exponent of convergence of poles less thanρlog(f) – .Set
F(z) =f(qz+c)n+f(z) +f(z).
If n≥,then F(z) –a has infinitely many zeros,where a∈C.
Some more general differential-q-shift-difference polynomials are investigated in the following.
Theorem . Let f(z)be a transcendental meromorphic function of finite logarithmic or-derρlog(f),with the logarithmic exponent of convergence of poles less thanρlog(f) – .Set
F(z) =f(z)mf(qz+c)nf(z).
If m,n satisfy m≥n+ or n≥m+ ,then F(z) –a(z)has infinitely many zeros.
Let
Pn(z) =anzn+an–zn–+· · ·+az+a
be a non-zero polynomial, wherea,a, . . . ,an(= ) are complex constants andtnis the
number of the distinct zeros ofPn(z). Then we also obtain the following results.
Theorem . Let f(z)be a transcendental meromorphic function of finite logarithmic or-derρlog(f),with the logarithmic exponent of convergence of poles less thanρlog(f) – .Set
F(z) =f(z)mPn
f(qz+c)
k
j= f(j)(z).
Theorem . Let f(z)be a transcendental meromorphic function of finite logarithmic or-derρlog(f),with the logarithmic exponent of convergence of poles less thanρlog(f) – .Set
F(z) =Pm
f(z)f(qz+c)n
k
j= f(j)(z).
If m≥n+k+ ,then F(z) –a(z)has infinitely many zeros.
Next, we investigate the uniqueness of differential-q-shift-difference polynomials of en-tire functions of order zero and obtain the following results.
Theorem . Let f(z)and g(z)be transcendental entire functions of order zero and n≥. If f(qz+c)nf(z)and g(qz+c)ng(z)share a non-zero polynomial p(z)CM, then f(qz+
c)nf(z) =g(qz+c)ng(z).
Theorem . Let f(z)and g(z)be transcendental entire functions of order zero and m≥ n+ tn+ .If f(z)mPn(f(qz+c))f(z)and g(z)mPn(g(qz+c))g(z)share a non-zero polynomial
p(z)CM,then f(z)mP
n(f(qz+c))f(z) =g(z)mPn(g(qz+c))g(z).
Theorem . Let f(z) and g(z) be transcendental entire functions of order zero and n≥m+ tm+ .If Pm(f(z))f(qz+c)nf(z)and Pm(g(z))g(qz+c)ng(z)share a non-zero
poly-nomial p(z)CM,then Pm(f(z))f(qz+c)nf(z) =Pm(g(z))g(qz+c)ng(z).
2 Some lemmas
To prove the above theorems, we need some lemmas as follows.
Lemma .[] Let f(z)be a non-constant meromorphic function and P(f) =a+af +
· · ·+anfn,where a,a, . . . ,anare complex constants and an= ,then
Tr,P(f)=nT(r,f) +S(r,f).
Lemma .[] Let f(z)be a transcendental meromorphic function of finite logarithmic order and q,ηbe two non-zero complex constants.Then we have
Tr,f(qz+η)=T(r,f) +S(r,f),
Nr,f(qz+η)=N(r,f) +S(r,f), N
r, f(qz+η)
=N
r, f
+S(r,f).
Lemma .[, Theorem .] Let f(z)be a non-constant zero-order meromorphic
func-tion and q∈C\ {}.Then
m
r,f(qz+η) f(z)
=S(r,f).
Lemma . [, p.] Let f(z)be a non-constant meromorphic function in the complex
plane and l be a positive integer.Then
Lemma .[] If f(z)is a transcendental meromorphic function of finite logarithmic order
ρlog(f),then for any two distinct small functions a(z)and b(z)with respect to f(z),we have
T(r,f)≤N
r, f–a
+N
r, f–b
+oU(r,f),
where U(r,f) = (logr)ρlog(f)is a logarithmic-type function of T(r,f).Furthermore,if T(r,f) has a finite lower logarithmic order
μ=lim inf
r→∞
logT(r,f) log logr ,
withρlog(f) –μ< ,then
T(r,f)≤N
r, f–a
+N
r, f–b
+oT(r,f).
Remark . From the proof of Lemma . (see [, Theorem .]), we can easily see that
complex valuesaandbcan be changed intoa(z) andb(z), wherea(z) andb(z) are two distinct small functions with respect tof(z).
Lemma . Let f(z)be a transcendental meromorphic function of order zero,F(z) =f(qz+ c)nf(z).Then we have
(n– )T(r,f) +S(r,f)≤(n– )T(r,f) –N(r,f) +S(r,f)
≤T(r,F)≤(n+ )T(r,f) +S(r,f). ()
Proof Iff(z) is a meromorphic function of order zero, from Lemmas . and ., we have
T(r,F)≤nT
r,f(qz+c)+Tr,f≤(n+ )T(r,f) +S(r,f).
On the other hand, from Lemmas . and . again, we have
(n+ )T(r,f) =Tr,f(qz+c)n++S(r,f)
≤T(r,F) +T
r,f(qz+c) f(z)
+S(r,f)
≤T(r,F) + T(r,f) +N(r,f) +S(r,f)
≤T(r,F) + T(r,f) +S(r,f).
Thus, we get ().
Lemma . Let f(z) be a transcendental meromorphic function of zero order, F(z) = f(z)mf(qz+c)nf(z).Then we have
and
T(r,F)≥ |m–n|T(r,f) –N(r,f) +S(r,f)≥
|m–n|– T(r,f) +S(r,f). ()
Proof Iff(z) is a meromorphic function of order zero, from Lemmas . and ., we have
T(r,F)≤mT(r,f) +nTr,f(qz+c)+Tr,f≤(m+n+ )T(r,f) +S(r,f), that is, we have (). On the other hand, from Lemmas . and ., we have
(n+m+ )T(r,f) =Tr,fn+m+=T
r, f(z)
n+F (z) f(qz+c)nf(z)
≤T(r,F) +T
r,f f
+T
r, f(z)
n
f(qz+c)n
≤T(r,F) + (n+ )T(r,f) +N(r,f) +S(r,f)
≤T(r,F) + (n+ )T(r,f) +S(r,f),
that is, we have (), where we assumem≥nwithout loss of generality.
Lemma . Let f(z) be a transcendental meromorphic function of order zero, F(z) = f(z)mP
n(f(qz+c)) jk=f(j)(z).Then we have
(m–n–k)T(r,f)≤T(r,F) + k(k+ )
N(r,f) +S(r,f),
T(r,F)≤
m+n+k(k+ )
T(r,f) +S(r,f).
Proof Sincef(z) is a transcendental meromorphic function of order zero, by Lemmas ., ., and ., we can easily get the second inequality. On the other hand, it follows by Lem-mas ., ., and . that
(m+k)T(r,f) =Tr,fm+k≤T
r, f(z)
kF(z)
Pn(f(qz+c)) kj=f(j)(z)
≤T(r,F) +T
r,Pn
f(qz+c)+T
r,
k
j= f(j)
f
≤T(r,F) + (n+ k)T(r,f) +k(k+ )
N(r,f) +S(r,f).
Thus, this completes the proof of Lemma ..
Similar to Lemma ., we have the following lemma.
Lemma . Let f(z) be a transcendental meromorphic function of zero order, F(z) = Pm(f(z))f(qz+c)n kj=f(j)(z).Then we have
(n–m–k)T(r,f)≤T(r,F) +k(k+ )
N(r,f) +S(r,f),
T(r,F)≤
m+n+k(k+ )
3 Proofs of Theorems 1.4-1.8
3.1 Proof of Theorem 1.4
It follows by Lemma . thatT(r,F) =O(T(r,f)) holds for allron a set of logarithmic density . Sincef(z) is transcendental andn≥,F(z) is transcendental by Lemma . again. Since the logarithmic exponent of convergence of poles off(z) less thanρlog(f) – , we have
lim sup
r→∞
logN(r,f)
log logr <ρlog(f).
Assume thatF(z) –a(z) has only finitely many zeros. Thus, by Lemmas ., .-., we have
(n– )T(r,f)≤T(r,F) +S(r,f)
≤N(r,F) +N
r, F–a
+oU(r,f)+S(r,f)
=Nr,f(qz+c)nf(z)+N
r, F–a
+oU(r,f)+S(r,f)
≤(n+ )N(r,f) +N
r, F–a
+oU(r,f)+S(r,f).
Thus, it follows that
(n– )T(r,f)≤(n+ )N(r,f) +oU(r,f)+S(r,f).
Sincen≥, the above inequality implies
lim sup
r→∞
logT(r,f)
log logr ≤lim supr→∞
logN(r,f)
log logr <ρlog(f),
which contradicts the fact thatT(r,f) has finite logarithmic orderρlog(f). Thus,F(z) –a(z) has infinitely many zeros, that is,f(qz+c)nf(z) –a(z) has infinitely many zeros.
This completes the proof of Theorem ..
3.2 Proof of Theorem 1.5
Sincef(z) is a transcendental meromorphic function of finite logarithmic order, we first claim thatf(z) +f(z) –a ≡. In fact, iff(z) +f(z) –a≡, that is,ff(z()–z)a ≡–. By solving the above equation, we havef(z) =Ae–z+a, whereAis a non-zero complex constant. Thus,
we haveρ(f) = , which contradicts the fact thatf(z) is of order zero. Thus, set
F(z) = a–f(z) –f
(z)
f(qz+c)n .
It follows by Lemmas . and . that
nT(r,f) =Tr,f(qz+c)n+S(r,f)
≤T
r, F(z) a–f(z) –f(z)
≤T(r,F) +T(r,f) +Tr,f+S(r,f)
≤T(r,F) + T(r,f) +N(r,f) +S(r,f),
that is,
(n– )T(r,f)≤T(r,F) +N(r,f) +S(r,f). ()
On the other hand, we can easily get
T(r,F)≤Tr,f(qz+c)n+T(r,f) +Tr,f+O()≤(n+ )T(r,f) +S(r,f). () And it follows from () andn≥ that
T(r,F) =OT(r,f)
holds for allron a set of logarithmic density . By Lemma ., we have
N
r, F–
=N
r, f(qz+c)
n
f(qz+c)n+f(z) +f(z) –a
≤N
r,
f(qz+c)n+f(z) +f(z) –a
+nN(r,f) +S(r,f).
Assume thatf(qz+c)n+f(z) +f(z) –ahas finitely many zeros, then
N
r, F–
≤nN(r,f) +S(r,f). ()
Since the logarithmic exponent of convergence of poles off(z) is less thanρlog(f) – , we have
lim sup
r→∞
logN(r,f)
log logr <ρlog(f).
Then, by Lemmas ., ., and (), we have
T(r,F)≤N
r, F
+N
r, F–
+oU(r,f)
≤N
r, a–f –f
+ nN(r,f) +S(r,f) +oU(r,f)
≤T(r,f) + (n+ )N(r,f) +S(r,f) +oU(r,f).
It follows by the above inequality and () that
(n– )T(r,f)≤(n+ )N(r,f) +S(r,f) +oU(r,f).
Sincen≥, the above inequality implies
lim sup
r→∞
logT(r,f)
log logr ≤lim supr→∞
logN(r,f)
which contradicts thatT(r,f) has finite logarithmic orderρlog(f). Thus,f(qz+c)n+f(z) + f(z) –ahas infinitely many zeros.
This completes the proof of Theorem ..
3.3 Proofs of Theorems 1.6, 1.7, and 1.8
Similar to the argument as in Theorem ., by applying Lemmas ., ., and . instead, we can easily prove Theorems ., ., and . respectively.
4 Proofs of Theorems 1.9-1.11
Here, we only give the proof of Theorem . because the methods of the proofs of The-orems ., ., and . are very similar.
4.1 Proof of Theorem 1.10
Denote
F(z) =f(z)mPn
f(qz+c)f(z), G(z) =g(z)mPn
g(qz+c)g(z).
Sincef(z) is a transcendental entire function of order zero, by Lemmas ., ., and ., we have
T(r,F)≤Tr,fm+T(r,Pn
f(qz+c)+Tr,f+O()
≤(n+m+ )T(r,f) +S(r,f), ()
and
(m+ )T(r,f) =Tr,fm+=T
r, f(z)F(z) Pn(f(qz+c))f(z)
≤T(r,F) +T
r, f f
+T
r,
Pn(f(qz+c))
+S(r,f)
≤T(r,F) + (n+ )T(r,f) +S(r,f). ()
Then it follows from () and () that
(m–n)T(r,f) +S(r,f)≤T(r,F)≤(n+m+ )T(r,f) +S(r,f). ()
We have by () thatS(r,F) =S(r,f). Similarly, we haveS(r,G) =S(r,g) and
(m–n)T(r,g) +S(r,g)≤T(r,G)≤(n+m+ )T(r,g) +S(r,g). ()
Sincef(z) andg(z) are entire functions of order zero and sharep(z)CM, we have
F(z) –p(z)
G(z) –p(z)=η, ()
whereηis a non-zero constant. Ifη= , then we haveF(z) =G(z), that is,f(z)mP n(f(qz+
c))f(z) =g(z)mP
Ifη= , then we have
F(z) –ηG(z) =p(z)( –η). ()
SincePn(z) hastndistinct zeros, by using the second main theorem and Lemma ., we
have
T(r,F)≤N(r,F) +N
r, F
+N
r,
F–p(z)( –η)
+S(r,F)
≤N
r,
Pn(f(qz+c))
+N
r, f
+N
r, f
+N
r, G
+S(r,f)
≤ tn j= N
r,
f(qz+c) –γj
+N
r, f
+N
r, f
+S(r,f)
+ tn j= N
r,
g(qz+c) –γj
+N
r, g
+N
r, g
+S(r,g)
≤(tn+ )T(r,f) + (tn+ )T(r,g) +S(r,f) +S(r,g), ()
whereγ,γ, . . . ,γtnare the distinct zeros ofPn(z). Similarly, we have
T(r,G)≤(tn+ )T(r,f) + (tn+ )T(r,g) +S(r,f) +S(r,g). ()
Then (), (), (), and () result in
(m–n)T(r,f) +T(r,g)≤(tn+ )
T(r,f) +T(r,g)+S(r,f) +S(r,g),
which contradictsm≥n+ tn+ .
This completes the proof of Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors drafted the manuscript, read, and approved the final manuscript.
Author details
1Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China.2Department of
Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, 333403, China.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11301233, 61202313, 11171119), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001) and Sponsored Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University of China, and the Foundation of Education Department of Jiangxi (GJJ14271, GJJ14644) of China.
Received: 12 April 2014 Accepted: 30 June 2014 Published:19 Aug 2014
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