Volume 2009, Article ID 416273,12pages doi:10.1155/2009/416273
Research Article
Subnormal Solutions of Second-Order
Nonhomogeneous Linear Differential Equations
with Periodic Coefficients
Zhi-Bo Huang,
1Zong-Xuan Chen,
1and Qian Li
21School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
2Department of Applied Mathematics, South China Agricultural University, Guangzhou 510640, China
Correspondence should be addressed to Zong-Xuan Chen,[email protected]
Received 8 February 2009; Accepted 24 May 2009
Recommended by Kunquan Lan
We obtain the representations of the subnormal solutions of nonhomogeneous linear diff er-ential equation f P1ez Q1e−zf P2ez Q2e−zf R1ez R2e−z, where
P1z, P2z, Q1z, Q2z, R1z,andR2zare polynomials inzsuch thatP1z, P2z, Q1z,and
Q2z are not all constants, degP1 > degP2. We partly resolve the question raised by G. G.
Gundersen and E. M. Steinbart in 1994.
Copyrightq2009 Zhi-Bo Huang 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.
1. Introduction
We use the standard notations from Nevanlinna theory in this papersee1–3.
The study of the properties of the solutions of a linear differential equation with periodic coefficients is one of the difficult aspects in the complex oscillation theory of differential equations. However, it is also one of the important aspects since it relates to many special functions. Some important researches were done by different authors; see, for instance,4–9.
Now, we firstly consider the second-order homogeneous linear differential equations
fP1ezfP2ezf 0, 1.1
Letfzbe an entire function. We define
ρe
f lim r→∞
logTr, f
r , 1.2
to be thee-type order offz.
Iffz/≡0 is a solution of1.1and iffzsatisfiesρef 0 , then we say thatfz is a subnormal solution of1.1. For convenience, we also say thatfz ≡ 0 is a subnormal solution of1.1.
H. Wittich has given the general forms of all subnormal solutions of 1.1 that are shown in the following theorem.
Theorem A see 9. If fz/≡0 is a subnormal solution of 1.1, where P1z and P2z are polynomials inzand are not both constants, thenfzmust have the form
fz ecza0a1ez· · ·amemz, 1.3
wherem≥0is an integer andc, a0, a1, . . . , amare constants witha0am/0.
G. G. Gundersen and E. M. Steinbart refined TheoremAand obtained the exact forms of subnormal solutions of1.1as follows.
Theorem Bsee6. In addition to the statement of TheoremA, the following statements hold with regard to the subnormal solutionsfzof 1.1.
iIfdegP1 >degP2 andP2/≡0,then any subnormal solutionfz/≡0of 1.1must have the form
fz
m
k0
ake−kz, 1.4
wherem≥1is an integer anda0, a1, . . . , amare constants witha0am/0.
iiIfP2≡0anddegP1 ≥1, then any subnormal solutionfzof1.1must be a constant.
iiiIfdegP1<degP2, then the only subnormal solutionfzof 1.1isfz≡0.
Whether the conclusions of Theorem A and Theorem B can be generalized or not, Gundersen and Steinbart considered the second-order nonhomogeneous linear differential equations
fP1ezfP2ezf R1ez R2
e−z, 1.5
In6, they also have raised the following problem, that is, what about the forms of the subnormal solutions of the equation
fP1ez Q1
e−zfP2ez Q2
e−zfR1ez R2
e−z, 1.6
whereP1z, P2z, Q1z, Q2z, R1z, andR2zare polynomials inzsuch thatP1z,P2z,
Q1z, andQ2zare not all constants?
In7, we have obtained the exact forms of all subnormal solutions of homogeneous equation
fP1ez Q1
e−zfP
2ez Q2
e−zf0, 1.7
whereP1z, P2z, Q1z, andQ2zare polynomials inzand are not all constants.
In this paper, we obtain the forms of subnormal solutions of nonhomogeneous linear differential equation1.6when degP1 >degP2. We have the following theorem.
Theorem 1.1. Suppose that fz is a subnormal solution of 1.6, where P1z, P2z,
Q1z, Q2z, R1z, andR2zare polynomials inzsuch thatP1z, P2z, Q1zand Q2zare not all constants.
iIfdegP1>degP2anddegP1>degR1, thenfzmust have the form
fz eβzg1ez g2
e−z, 1.8
whereβis a constant,g1zandg2zare polynomials inz.
iiIfdegP1>degP2anddegP1≤degR1, thenfzmust have the form
fz eβzg1ez g2
e−zc1zg3
e−zc2g4
e−zg0ez, 1.9
where β is a constant, c1 and c2 are constants that may or may not be equal to zero,
g0zmay be equal to zero or may be a polynomial inz,g1z, g2z, g3z, andg4z are polynomials inzwithdeg{g3} ≥1.
2. Lemmas for the Proof
In order to proveTheorem 1.1, we need some lemmas.
Lemma 2.1see7. Suppose thatfzis a subnormal solution of1.7, whereP1z,P2z,Q1z andQ2zare polynomials inzand are not all constants.
iIfdegP1>degP2andP2Q2≡0,then any subnormal solutionfzmust be a constant.
iiIfdegP1>degP2andP2Q2/≡0,thenfz/≡0must have the form
fz g2
e−z, 2.1
Lemma 2.2see10. Letfzbe a transcendental meromorphic function, letα >1be a given real constant, and letk > j ≥0. Then there exists a constantCCα> 0such that the following two statements hold (wherer|z|).
iThere exists a setE1⊂−π, πthat has linear measure zero such that ifψ∈−π, π\E1, then there is a constantRRψ>0such that for allzsatisfyingargzψand|z| ≥R,
one has
fkz
fjz ≤C
Tαr, f r log
αrlogTαr, f k−j
. 2.2
iiThere exists a setE2 ⊂0,∞that has finite logarithmic measure such that2.2holds for allzsatisfying|z|r /∈E2∪0,1.
Lemma 2.3 see6. Let 0 < r1 < r2 < · · · < rj < · · · with rj → ∞as j → ∞, and let
θ1< θ2< θ12π.Let
W1
z: argzθ1
, W2
z: argzθ2
,
W3
z:|z|rj for some j andθ1≤argz≤θ2
, 2.3
and set
W W1∪W2∪W3. 2.4
Letfzbe analytic on the setW. Suppose thatfzis unbounded on the setW. Then there exists an infinite sequence of pointszj ∈Wwith|zj| → ∞asj → ∞such that
fzj
−→ ∞,
fzj
fzj
≥1o1 1 8zj
.
2.5
Lemma 2.4see8. Consider the nth- order differential equation of the form
P0
ez, e−zfnP1
ez, e−zfn−1· · ·Pn
ez, e−zfPn1
ez, e−z, 2.6
wherePjez, e−z j0,1,2, . . . , n1are polynomials inezande−zwithP0ez, e−z/0. Suppose that fzis an entire and subnormal solution of 2.6 and thatfz can be expressed asfz
eβzGez, whereβis a constant andGζis analytic on0<|ζ|<∞. Thenfzhas the form
fz eβzg1ez g2
e−z, 2.7
As an application ofLemma 2.4, one has the following lemma.
Lemma 2.5. Suppose that fz is an entire subnormal solution of 2.6, where Pjez, e−zj 0,1,2, . . . , n1are polynomials inezande−zwithP
0ez, e−z/0, and thatfzandfz2πi are linearly dependent. Thenfzhas the form
fz eβzg1ez g2
e−z, 2.8
whereβis a constant andg1zandg2zare polynomials inz.
Proof. Sincefzis entire and is linearly dependent withfz2πi,fzcan be written as
fz eβzGez see11, page 382, whereβis a constant andGζis analytic on 0<|ζ|<∞. Then we have the representation fromLemma 2.4.
Lemma 2.6. Suppose thatfzis a solution of 1.6, whereP1z,P2z,Q1z,Q2z,R1z, and
R2zare polynomials inzsuch thatP1z,P2z,Q1zandQ2zare not all constants. If
degP2<degP1<degR1, 2.9
then there exists a polynomialg0zsuch that
fz gz g0ez, 2.10
wheregzis a solution of
gP1ez Q1
e−zgP2ez Q2
e−zgR3ez R4
e−z, 2.11
whereR3zandR4zare polynomials inzwithdegR3≤degP1.
Proof. LetndegR1−degP1≥1,and set
gz fz−aenz, 2.12
whereais the constant such that
deg{R1z−anznP1z}<degR1. 2.13
It follows from1.6and2.12that
gP1ez Q1
e−zgP
2ez Q2
e−zgT
1ez T2
e−z, 2.14
where
T1ez T2
e−zR1ez R2
e−z−anenzP1ez Q1
e−z
−aenzP2ez Q2
e−z−an2enz.
SoT1ezand T2e−z are polynomials inez and e−z, respectively, and degT1 < degR1 by
2.13, butT1ezandT2e−zhave the exact representations that depend on the relations of
n,degQ1, and degQ2. If degT1 ≤ degP1, then2.14is of the form2.11, and2.12gives
2.10. If degT1>degP1, then we repeat the above process finite times until we obtain2.10 and2.11. This completes the proof ofLemma 2.6.
3. Proof of Theorem
In this section, we will proveTheorem 1.1.
Proof. iSuppose thatfzis a subnormal solution of1.6with degP1>degP2and degP1> degR1. Iffzis a polynomial solution of1.6, thenfzmust be a constant, which is of the form1.8. Thus we suppose thatfzis transcendental. It follows fromLemma 2.2ithat there exists a set E1 ⊂ −π, πthat has linear measure zero such that ifψ ∈ −π, π\E1, then there is a constantR Rψ>0 such that for allzsatisfying argz ψand|z| ≥R, we have
ff
≤CT
2r, f
r log
2rlogT2r, f, 3.1
whereC >0 is a constant andr |z|. It also follows fromLemma 2.2iithat there exists a setE2 ⊂ 0,∞that has finite logarithmic measure such that3.1holds for allzsatisfying
|z|∈E2∪0,1.
Now letr1, r2, . . . , rj,be an infinite sequence satisfying 1< r1< r2 <· · ·< rj <· · · such thatrj∈E 2for alljandrj → ∞asj → ∞.Letε0be a small constant such that−π/2ε0/∈E1 andπ/2−ε0/∈E1. Set
W1
z: argz−π
2 ε0
, W2
z: argz π
2 −ε0
,
W3
z:|z|rj for somej and−
π
2 ε0≤argz≤
π
2 −ε0
,
3.2
and set
WW1∪W2∪W3. 3.3
From above, we have that3.1holds on the setW.
We now assert thatfzis bounded on the setW. On the contrary, it follows from
Lemma 2.3that there exists a sequence of pointszj ∈Wwith|zj| → ∞asj → ∞such that fz
j−→ ∞, 3.4
fzj
fzj
≥1o1 1 8zj
By1.6, we have for allzj∈W,
fzj
fzj
· 1
P1ezj Q1e−zj
1 P2e zj Q
2e−zj
P1ezj Q1e−zj·
fzj
fzj
1
fzj
·R1ezj R2e−zj
P1ezj Q1e−zj
.
3.6
It follows from3.4–3.6andρef 0 that3.6yields 1≡0 as|zj| → ∞on the setW.This is a contradiction.
By the maximum modulus principle,fzis bounded in the angular domain
Dz:−π
2 ε0≤argz≤
π
2 −ε0
. 3.7
However, we know
fz fz0 z
z0
ftdt, 3.8
where the integral offtis defined on the simple contourC, extending from a pointz0to a pointzin the complex domain.
So we obtain
fzO|z|, 3.9
asz → ∞in the angular domainD.
Thus , from the Cauchy integral formula, we obtain
fzO1, 3.10
asz → ∞ in the angular domainD. By1.6,3.8, and 3.9, we have for some constant
A >0
fz≤exp{−1o1A|z|}, 3.11
asz → ∞in the angular domainD. Together with3.8and3.11,fzis bounded in the angular domainD.
Iffz≡fz2πi, it follows fromLemma 2.5thatfzmust have the form1.8. Iffz/≡fz2πi, sincefzis a subnormal solution of1.6, so isfz2πi. Thus,
hz fz−fz2πi 3.12
Case 1. IfP2Q2≡0,we have, byLemma 2.1i,
hz C, 3.13
whereCis a constant. Hencehz≡0, that is,
fz≡fz2πi. 3.14
From this,fzcan be written asfz eβ1zG
1ezsee11, page 382, whereβ1is a constant andG1zis analytic on 0<|z|<∞. Thus,fzcan be written asfz eβ2zG2ez, whereβ2 is a constant andG2zis analytic on 0<|z|<∞. It follows fromLemma 2.4that
fz eβzg1ez g2
e−z, 3.15
whereβis a constant,g1zandg2zare polynomials inz. Thus,fzhas the form of1.8.
Case 2. IfP2Q2≡/0,we obtain fromLemma 2.1iithat
hz fz−fz2πi g3
e−z, 3.16
whereg3zis a polynomial inzwith deg{g3} ≥1.
However, we can assert thathz g3e−z≡0 in3.16. Otherwise, there existsz0∈D such that
g3
e−za /0. 3.17
By3.16, we have
fz2πi−fz4πi g3
e−z. 3.18
Thus from3.16and3.18, we have
fz−fz4πi 2g3
e−z. 3.19
By repeating this process finite times, we obtain that for any integern≥1,
fz−fzn·2πi ng3
e−z. 3.20
We have, by3.17and3.20,
asn → ∞.This is a contradiction to the fact thatfzis bounded in the angular domainD. This shows thatP2Q2/≡0 is not possible whenfz/≡fz2πiunder the hypotheses. This completes the proof of parti.
iiWe firstly suppose that degP1degR1. Sincefzis a subnormal solution of1.6, so isfz2πi. Set
hz fz−fz2πi. 3.22
Thenhzis a subnormal solution of1.7. Now ifhz≡0, this shows thatfz≡fz2πi andfzhas the form of1.9byLemma 2.5. Thus, we suppose thathz/≡0 in the following.
Now, assume that degP1>degP2.
IfP2Q2 ≡ 0,it follows from the proof of Case1ofTheorem 1.1ithatfzhas the form of1.9.
IfP2Q2/≡0,we obtain fromLemma 2.1iithat
hz fz−fz2πi g3
e−z, 3.23
whereg3zis a polynomial inzwith deg{g3} ≥1. Set
g3
e−z
m
k0
ake−kz, 3.24
wherem≥1 is an integer andakk0,1, . . . , mare constants witham/0. LetndegP1degR1and set
P1ez n
k0
pkekz, R1ez n
k0
rkekz, 3.25
wherepnrn/0.
Now, we will discuss the following two cases.
Case A. We considera0/0 in3.24. Letc1be a constant defined by
c1a0pnrn, 3.26
and set
H1z c1zhz. 3.27
Sincehzis a subnormal solution of1.7, it follows from3.27thatH1zsatisfies
H1P1ez Q1
e−zH1P2ez Q2
e−zH12c1hz c1hz
P1ez Q1
We obtain from3.23–3.28that
H1P1ez Q1
e−zH1P2ez Q2
e−zH1rnenzT3ez T4
e−z, 3.29
whereT3zandT4zare polynomials inzwith degT3≤n−1. Set
φ1z H1z−fz. 3.30
It follows from 1.6, 3.25, 3.29 and 3.30 that φ1 P1ez Q1e−zφ1 P2ez
Q2e−zφ1T3ez T4e−z− n−1
k0
rkekz−R2e−z.
Set
S1ez T3ez− n−1
k0
rkekz, S2
e−zT4
e−z−R2
e−z. 3.31
So degS1≤n−1<degP1, andφ1zsatisfies
φ1P1ez Q1
e−zφ 1
P2ez Q2
e−zφ
1 S1ez S2
e−z. 3.32
We have by3.27thathz is a subnormal solution of1.7,H1zis a subnormal solution of3.29. Moreover,φ1zis also a subnormal solution of3.32by3.30andfzis a subnormal solution of1.6. Thus, we deduce fromTheorem 1.1iand3.32with degP1> degS1thatφ1zhas the form
φ1z eβz
g1ez g2
e−z, 3.33
whereβis a constant,g1zandg2zare polynomials inz. Hence3.23,3.24,3.27,3.30, and3.33yield
fz eβz−g1ez−g2
e−zc1zg3
e−z, 3.34
wherec1/0 andβare constants,g1z, g2z, andg3zare polynomials inzwith degg3≥1. This is the form of1.9.
Case B. We considera00 in3.24. Letc2be a constant defined by
c2saspn rn, 3.35
wheres ∈ {0,1, . . . , m}is a number such thatasis the first coefficienta0, a1, . . . , amin3.24 which is not equal to zero. Set
Similar to the proof of CaseAofTheorem 1.1ii, we have
fz eβz−g1ez−g2
e−zc2
eszg3
e−z, 3.37
wherec2/0 andβare constants,g1z, g2zand,g3zare polynomials inzwith degg3≥1. Setg4e−z c2eszg3e−z. Theng4e−z/≡0 is a polynomial ine−zby the hypotheses ofsin
3.36. This is the form of1.9. We have provedTheorem 1.1iiwhen degP1degR1.
Now we suppose that degP1<degR1. ByLemma 2.6, there exists a polynomialg0z inz, satisfies2.10and2.11.
Since degR3≤degP1and since we have provedTheorem 1.1holds in the cases when degP1≥degR1holds, we can apply this result to2.11.
If degP1 >degR3, it follows fromTheorem 1.1ithat
gz eβzg1ez g2
e−z, 3.38
whereβis a constant,g1zandg2zare polynomials inz. By2.10and3.38, we obtain that
fz g0ez eβz
g1ez g2
e−z, 3.39
whereβis a constant,g0z, g1zandg2zare polynomials inz. This is a form of1.9. If degP1 degR3, it follows from the proof ofTheorem 1.1iiwhen degP1 degR1 that
gz eβzg1ez g2
e−zc1zg3
e−zc2g4
e−z, 3.40
where g1z, g2z, g3z and g4z are polynomials in z with degg3 ≥ 1, c1 and c2 are constants that may or may not be equal to zero. By2.10and 3.40, we obtain thatfz
has the form of1.9.Theorem 1.1iiis completed.
Now, we give some examples to show thatTheorem 1.1is correct.
Example 3.1. Letfz e−ze−2z, thenfzsatisfies
fe2ze−z1f−ez2e−z−1f −ez−32e−2z. 3.41
This is an example ofTheorem 1.1i.
Example 3.2. Letfz z1e−z e−2z, thenfzsatisfies
fe2z2eze−z3feze−z1f e2z3ez3−e−ze−3z. 3.42
Example 3.3. Letfz ze−z eze−z, thenfzsatisfies
fe2zeze−zfeze−zfe3z2e2zez−e−ze−2z3. 3.43
This is an example ofTheorem 1.1iiwith degP1>degP2and degP1<degR1.
Acknowledgements
The authors are very grateful to the referee for his her many valuable comments and suggestions which greatly improved the presentation of this paper. The project was supposed by the National Natural Science Foundation of China no. 10871076, and also partly supposed by the School of Mathematical Sciences Foundation of SCNU, China.
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