R E S E A R C H
Open Access
Existence and properties of meromorphic
solutions of some
q
-difference equations
Na Xu
*and Chun-Ping Zhong
*Correspondence:
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P.R. China
Abstract
In this paper, we investigate the existence and growth of solutions of theq-difference equationni=1f(qiz) =R(z,f(z)), whereR(z,f(z)) is an irreducible rational function in f(z). We also give an estimation of the growth of transcendental meromorphic solutions of the equationni=1f(qiz) =f(z)m.
MSC: 30D35; 39A05
Keywords: q-difference equation; meromorphic solution; growth
1 Introduction and main results
A meromorphic functionf(z) means meromorphic in the complex planeC. If no poles occur, thenf(z) reduces to an entire function. Assume thatn(r,f) counts the number of poles off in|z| ≤r, each pole is counted according to its multiplicity, and thatn(r,f) counts the number of the distinct poles off in|z| ≤r, ignoring the multiplicity. The characteristic function off is defined by
T(r,f) :=m(r,f) +N(r,f),
where
N(r,f) :=
r
n(t,f) –n(,f)
t dt+n(,f)logr
and
m(r,f) := π
π
log+freiθdθ.
For more notations and definitions of the Nevanlinna value distribution theory of mero-morphic functions, we refer to [, ].
A meromorphic functionα(z) is called a small function with respect tof(z), ifT(r,α) =
S(r,f), whereS(r,f) denotes any quantity satisfyingS(r,f) =o(T(r,f)) asr→ ∞outside a possible exceptional setEof logarithmic density . The order and the exponent of con-vergence of zeros of meromorphic functionf(z) is, respectively, defined as
σ(f) =lim sup
r→∞
logT(r,f)
logr , λ(f) =lim supr→∞
logN(r,f)
logr .
The difference operators for a meromorphic functionf(z) are defined as
cf(z) =f(z+c) –f(z) (c= ),
∇qf(z) =f(qz) –f(z) (q= , ).
In the following,f(qz+c) is theq-shift off(z),f(qz) –f(z) is theq-difference off(z), where
q= , . If an equation includesq-shifts orq-differences off(z), then the equation is called theq-difference equation. A Borel exceptional value off(z) is any valueasatisfyingλ(f –
a) <σ(f).
In the last two decades, the existence and growth of meromorphic solutions of differ-ence equations have been investigated in many papers [–]. Recently, with the develop-ment of theq-difference analog of Nevanlinna theory, there has been a renewed interest in studying meromorphic solutions ofq-difference equations. For instance, Zheng and Chen [] considered the growth problem of transcendental meromorphic solutions of someq -difference equations.
Theorem A[, Theorem ] Suppose that f is a transcendental meromorphic solution of the equation
n
j=
aj(z)f
qjz=Rz,f(z)=P(z,f(z))
Q(z,f(z)),
where q∈C,|q|> ,the coefficients aj(z)are rational functions and P,Q are relatively prime
polynomials in f over the rational functions satisfying p=degfP,t=degfQ,d=p–t≥.
If f has infinitely many poles,then for sufficiently large r,n(r,f)≥Kd
logr
nlog|q|holds for some
constant K> .Thus,the lower order of f,which has infinitely many poles,satisfiesμ(f)≥
d
logd nlog|q|.
In [], Heittokangaset al.first considered meromorphic solutions with Borel excep-tional zeros and poles of some type of difference equations and obtained the result as follows.
Theorem B[, Theorem ] Let c, . . . ,cn∈C\ {}and suppose that f is a non-rational
meromorphic solution of a difference equation of the form
n
i=
f(z+ci) =
a(z) +a(z)f(z) +· · ·+ap(z)f(z)p
b(z) +b(z)f(z) +· · ·+bt(z)f(z)t
,
with meromorphic coefficients ai(z),bj(z)of growth S(r,f)such that ap(z)bt(z)≡.If
max λ(f),λ
f
<ρ(f),
then the above equation is of the form
n
i=
f(z+ci) =c(z)f(z)k,
Recently, Zheng and Chen [] considered aq-difference equation under a condition sim-ilar to Theorem B on meromorphic solution, and they obtained the following result.
Theorem C[, Theorem ] Suppose that f is a transcendental meromorphic solution of a q-difference equation of the form
n
i=
f(qiz) =R
z,f(z)=a(z) +a(z)f(z) +· · ·+ap(z)f(z)
p
b(z) +b(z)f(z) +· · ·+bt(z)f(z)t
, (.)
where qi∈C\ {, },i= , . . . ,n,and R(z,f)is an irreducible rational function in f with
meromorphic coefficients ai(z) (i= , . . . ,p)and bj(z) (j= , . . . ,t)of growth S(r,f)such that
bt(z)≡,ap(z)≡.If
max λ(f),λ
f
<ρ(f) =ρ,
then the above equation is reduced to the form
n
i=
f(qiz) =ap(z)f(z)p
or
n
i=
f(qiz) =
a(z)
f(z)t.
For theq-difference equation (.), Theorem C only considered the case when solutions have Borel exceptional zeros and poles. But how about the existence and growth of mero-morphic solutions of (.)? Theorem . considers under what conditions (.) will not have solutions with zero order.
Theorem . Let R(z,f)be an irreducible rational function in f with meromorphic coeffi-cients ai(z) (i= , . . . ,p)and bj(z) (j= , . . . ,t)of growth S(r,f),d=max{p,t},qi∈C\ {, },
i= , . . . ,n.
() Ifd>n,then(.)has no transcendental meromorphic solution of zero order. () Ifd=n,then(.)has no transcendental entire solution of zero order.
As many papers (see [, ]) obtained the lower bound of the order of solutions of dif-ference equations. The natural question arises of the upper bound of the order of the so-lutions of (.). The following theorem answers this question partly.
Theorem . Suppose that f(z)is a transcendental meromorphic solution of (.),where R(z,f) is an irreducible rational function in f with meromorphic coefficients ai(z) (i=
, . . . ,p)and bj(z) (j= , . . . ,t)of growth S(r,f),d=max{p,t},qi∈C,|qi|> (i= , . . . ,n), |q| ≤ |q| ≤ · · · ≤ |qn–|<|qn|.Then
When the functionR(z,f) in (.) is reduced tof(z)m, we consider theq-difference
equa-tion
n
i=
f(qiz) =f(z)m. (.)
From the proof of Theorem ., one can immediately get the following corollary.
Corollary . Let m,n be positive integers,qi∈C\ {, },i= , . . . ,n.
() Ifm>n,then(.)has no transcendental meromorphic solution of zero order. () Ifm=n,then(.)has no transcendental entire solution of zero order.
The following theorem gives an estimation of the growth of the meromorphic solutions of (.), whereqi∈C,|qi|> (i= , . . . ,n).
Theorem . Suppose that f(z)is a transcendental meromorphic solution of (.),where m,n are positive integers,|qi|> (i= , . . . ,n),|q|=max{|q|,|q|, . . . ,|qn|}.Then
σ(f)≥μ(f)≥logm–logn log|q| .
Remark If|qk|=|q|, we denote|s|=max{|qi|,i= , . . . ,n,i=k}, then|s| ≤ |q|. If|s| =|q|, then from the proof of Theorem ., we can immediately get
σ(f)≤ log(m+n– ) log|q|–log|s|.
In the following, we consider transcendental entire solutions withλ(f) <σ(f) of (.).
Theorem . Suppose that f(z)is a transcendental entire solution of finite order of (.),
where m,n are positive integers,qi∈C\ {, },i= , . . . ,n,ifλ(f) <σ(f),then
n
i=
qσi(f)=m.
The following example shows that the case of Theorem . can occur.
Example Letm= ,q= ,q= , then the transcendental entire functionf(z) =ez
satisfies the equation
f(z)f(z) =f(z).
Here,f(z) has finitely many zeros and satisfiesqσ(f)+qσ(f)=m.
2 Lemmas
Lemma .[] (Valiron-Mohon‘ko) Let f(z)be a meromorphic function,then for all ir-reducible rational functions in f,
Rz,f(z)=
m
i=ai(z)f(z)i
n
j=bj(z)f(z)j
,
with meromorphic coefficients ai(z),bj(z),the characteristic function of R(z,f(z))satisfies
Tr,Rz,f(z)=dT(r,f) +Oψ(r),
where d=max{m,n}andψ(r) =max{T(r,ai),T(r,bj)}.
Lemma . gives us the relationship of the characteristic function between f(z) and
f(qz), provided thatf(z) is a non-constant meromorphic function of zero order.
Lemma .[] If f(z)is a non-constant meromorphic function of zero order,and q∈C\
{},then
Tr,f(qz)= +o()T(r,f) (.)
on a set of lower logarithmic density.
Lemma .[] Let f(z)be a non-constant meromorphic function of zero order,and q∈ C\ {},then
m
r,f(qz)
f(z)
=S(r,f) (.)
on a set of logarithmic density.
The next lemma is the relationship betweenT(r,f(qz)) andT(|q|r,f(z)).
Lemma .[] Let f(z)be a meromorphic function,and let q∈C\ {},then
Tr,f(qz)=T|q|r,f+O(). (.)
Lemma .[] Letφ: (,∞)→(,∞)be a monotone increasing function,and let f be a non-constant meromorphic function.If for some real constantα∈(, ),there exist real constants K> and K≥such that
T(r,f)≤Kφ(αr) +KT(αr,f) +S(αr,f),
then
σ(f)≤ logK
–logα+lim supr→∞ logφ(r)
logr .
Lemma . [] If an entire function f has a finite exponent of convergenceλ(f)for its zero-sequence,then f has a representation in the form
f(z) =Q(z)eg(z),
satisfyingλ(Q) =σ(Q) =λ(f).Further,if f is of finite order,then g in the above form is a polynomial of degree less than or equal to the order of f.
3 The proofs
3.1 Proof of Theorem 1.1
() Suppose thatf(z) is a transcendental meromorphic solution of zero order of (.), it follows from Lemma . that
on a set of lower logarithmic density . It is clear that (.) is a contradiction whend>n. Thus ifd>n, (.) has no transcendental meromorphic solution of zero order.
() Suppose thatf(z) is a transcendental entire solution of zero order of (.), it follows from (.) that
Sincef(z) is a transcendental entire solution of zero order, by Lemma . and Lemma ., the above inequality can be reduced to
nT(r,f)≤dT(r,f) +S(r,f) (.)
on a set of lower logarithmic density . It is clear that (.) is a contradiction whend<n. Using the same method of the proof of (), we can get ifd>n, (.) has no transcendental entire solution of zero order.
3.2 Proof of Theorem 1.2
It follows from (.) and Lemma . that
Tr,f(qnz)
=T
r,n–R(z,f)
i= f(qiz)
≤dT(r,f) +
n–
i=
Tr,f(qiz)
+S(r,f).
Since|qi|> (i= , . . . ,n),|q| ≤ |q| ≤ · · · ≤ |qn–|<|qn|, by Lemma ., we obtain
T|qn|r,f≤dT(r,f) + (n– )T|qn–|r,f+S|qn–|r,f.
SettingR=|qn|r,α=|qn–|
|qn| , we have
T(R,f)≤(d+n– )T(αR,f) +S(αR,f). (.)
By Lemma . and (.), we obtain
σ(f)≤ log(d+n– ) log|qn|–log|qn–|.
3.3 Proof of Theorem 1.4
We will divide the argument into two cases.
Case. Ifn≥m, it is easily to seeσ(f)≥μ(f)≥loglog|m–logq| nis obviously true.
Case. Ifn<m, by Corollary ., we get whenn<m, (.) has no transcendental mero-morphic solution of zero order. Sof(z) is a transcendental meromorphic solution of pos-itive order. It follows from Lemma . that
Tr,f(qiz)
=T|qi|r,f+O().
Note that|q|=max{|q|,|q|, . . . ,|qn|}, thus by (.), we obtain
mT(r,f) =Tr,fm=T
r,
n
i=
f(qiz)
≤ n
i=
Tr,f(qiz)
+S(r,f)
≤nT|q|r,f+S(r,f).
By the above inequality, we have, for any given( << ),
mT(r,f)≤n( +)T|q|r,f (.)
outside of a possible exceptional set of finite logarithmic measure. Now we can use [, Lemma ..] to deal with the exceptional set here. It follows from (.) that for any given
α> , there exists anr> such that
holds for allr≥r. This implies that
Tα|q|r,f≥ m
n( +)T(r,f). (.)
Inductively, for anyk∈N, we obtain
where
p(z) =g(qz)· · ·g(qnz)ead–(q
d–
+···+qdn–)zd–+···+na.
On the other hand, we have
f(z)m=g(z)memh(z)=g(z)memadzd+mad–zd–+···+ma. (.)
Sinceσ(g) <d, by (.), (.) and (.), we obtain
p(z)ead(qd+···+qdn)zd=g(z)memadzd+mad–zd–+···+ma,
which implies that
qd+· · ·+qdn=m.
Sinceσ(f) =deg(h(z)) =d, we have
n
i=
qσi(f)=m.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper.
Acknowledgements
This research was partly supported by National Natural Science Foundation of China (11271304); the Natural Science Foundation of Fujian Province of China for Distinguished Young Scholars (2013J06001); and the Program for New Century Excellent Talents in University (NCET-13-0510).
Received: 14 October 2014 Accepted: 25 December 2014
References
1. Hayman, W: Meromorphic Functions. Clarendon, Oxford (1964)
2. Yang, L: Value Distribution Theory. Springer, Berlin (1993); Science Press, Beijing (1982)
3. Barnett, DC, Halburd, RG, Korhonen, RJ, Morgan, W: Nevanlinna theory for theq-difference operator and meromorphic solutions ofq-difference equations. Proc. R. Soc. Edinb., Sect. A137, 457-474 (2007)
4. Bergweiler, W, Ishizaki, K, Yanagihara, N: Meromorphic solutions of some functional equations. Methods Appl. Anal. 5(3), 248-259 (1998). Correction: Methods Appl. Anal.6(4), 617-618 (1999)
5. Bergweiler, W, Langley, JK: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc.142, 133-147 (2007)
6. Yi, HX, Yang, CC: Uniqueness Theory of Meromorphic Functions. Science Press, Beijing (1995); Kluwer Academic, Dordrecht (2003)
7. Zhang, JL, Korhonen, RJ: On the Nevanlinna characteristic off(qz) and its applications. J. Math. Anal. Appl.369, 537-544 (2010)
8. Zheng, XM, Chen, ZX: Some properties of meromorphic solutions ofq-difference equations. J. Math. Anal. Appl. 361(2), 472-480 (2010)
9. Zheng, XM, Chen, ZX: On properties ofq-difference equations. Acta Math. Sci.32(2), 724-734 (2012)
10. Heittokangas, J, Korhonen, R, Laine, I, Rieppo, J, Tohge, K: Complex difference equations of Malmquist type. Comput. Methods Funct. Theory1(1), 27-39 (2001)
11. Zheng, XM, Tu, J: Growth of meromorphic solutions of linear difference equations. J. Math. Anal. Appl.384, 349-356 (2011)
12. Laine, I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993)
13. Gundersen, GG, Heittokangas, J, Laine, I, Rieppo, J, Yang, D: Meromorphic solutions of generalized Schröder equations. Aequ. Math.63(1-2), 110-135 (2002)
