Subnormal Solutions of Second-Order

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,

Zong-Xuan Chen,

and Qian Li

1School 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 differ- ential equation f P1e^z Q1e^−zf P2e^z Q2e^−zf R1e^z 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

fP1e^zfP2e^zf 0, 1.1

whereP1zandP2zare polynomials inzand are not both constants. It is well known that every solutionfzof1.1is an entire function.

Letfzbe an entire function. We define

ρ_e f

lim

r→∞

logT r, 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 e^cza0a1e^z· · ·ame^mz, 1.3 wherem≥0 is 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.

iIf degP1 >degP2 andP2/≡0,then any subnormal solutionfz/≡0 of 1.1must have the form

fz ^m

k0

a_ke^−kz, 1.4

wherem≥1 is an integer anda₀, a₁, . . . , a_mare constants witha₀a_m/0.

iiIfP2≡0 and degP1 ≥1, then any subnormal solutionfzof1.1must be a constant.

iiiIf degP₁<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

fP1e^zfP2e^zf R1e^z R2

e^−z

, 1.5

whereP₁z, P2z, R1z, andR₂zare polynomials inzsuch thatP₁z, P2zare not both constants. They found the exact forms of all subnormal solutions of1.5, that is, what is mentioned in6, Theorem 2.2, Theorem 2.3 and Theorem 2.4.

In6, they also have raised the following problem, that is, what about the forms of the subnormal solutions of the equation

f

P₁e^z Q₁ e^−z

f

P₂e^z Q₂ e^−z

fR₁e^z 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

f

P₁e^z Q₁ e^−z

f

P₂e^z Q₂ e^−z

f0, 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.

iIf degP₁>degP₂and degP₁>degR₁, thenfzmust have the form fz e^βz

g1e^z g2

e^−z

, 1.8

whereβis a constant,g1zandg2zare polynomials inz.

iiIf degP₁>degP₂and degP₁≤degR₁, thenfzmust have the form fz e^βz

g1e^z g2

e^−z c1zg3

e^−z c2g4

e^−z

g0e^z, 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 inzwith deg{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.

iIf degP₁>degP₂andP₂Q₂≡0,then any subnormal solutionfzmust be a constant.

iiIf degP₁>degP₂andP₂Q₂/≡0,thenfz/≡0 must have the form fz g2

e^−z

, 2.1

whereg₂zis a polynomial inzwith deg{g2} ≥1.

Lemma 2.2see10. Letfzbe a transcendental meromorphic function, letα >1 be a given real constant, and letk > j ≥0. Then there exists a constantCCα> 0 such 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>0 such that for allzsatisfying argzψand|z| ≥R, one has

f^kz f^jz ≤C

T αr, f

r log^αrlogT

αr, 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|r_j 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

f z_j

−→ ∞,

f

zj

f

zj

≥1o1 1

8zj. 2.5

Lemma 2.4see8. Consider the nth- order differential equation of the form P0

e^z, e^−z

fⁿP1

e^z, e^−z

fⁿ⁻¹· · ·Pn

e^z, e^−z

fP_n1 e^z, e^−z

, 2.6

whereP_je^z, e^−z j0,1,2, . . . , n1are polynomials ine^zande^−zwithP₀e^z, e^−z/0. Suppose that fzis an entire and subnormal solution of 2.6 and thatfz can be expressed asfz e^βzGe^z, whereβis a constant andGζis analytic on 0<|ζ|<∞. Thenfzhas the form

fz e^βz

g₁e^z g₂ e^−z

, 2.7

whereβis a constant andg₁zandg₂zare polynomials inz.

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 P_je^z, e^−zj 0,1,2, . . . , n1are polynomials ine^zande^−zwithP₀e^z, e^−z/0, and thatfzandfz2πi are linearly dependent. Thenfzhas the form

fz e^βz

g₁e^z g₂ e^−z

, 2.8

whereβis a constant andg1zandg2zare polynomials inz.

Proof. Sincefzis entire and is linearly dependent withfz2πi,fzcan be written as fz e^βzGe^z 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 g₀e^z, 2.10

wheregzis a solution of g

P₁e^z Q₁ e^−z

g

P₂e^z Q₂ e^−z

gR₃e^z R₄ e^−z

, 2.11

whereR3zandR4zare polynomials inzwith degR3≤degP1. Proof. LetndegR₁−degP₁≥1,and set

gz fz−ae^nz, 2.12

whereais the constant such that

deg{R1z−anzⁿP1z}<degR1. 2.13

It follows from1.6and2.12that g

P₁e^z Q₁ e^−z

g

P₂e^z Q₂ e^−z

gT₁e^z T₂ e^−z

, 2.14

where

T1e^z T2

e^−z

R1e^z R2

e^−z

−ane^nz

P1e^z Q1

e^−z

−ae^nz

P2e^z Q2

e^−z

−an²e^nz. 2.15

SoT1e^zand T2e^−z are polynomials ine^z and e^−z, respectively, and degT1 < degR1 by 2.13, butT₁e^zandT₂e^−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

f f

≤CT 2r, f

r log²rlogT 2r, 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 thatr_j∈E 2for alljandr_j → ∞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|r_j 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 pointsz_j ∈Wwith|zj| → ∞asj → ∞such that

f

zj−→ ∞, 3.4

f z_j f

z_j

≥1o1 1

8z_j. 3.5

By1.6, we have for allzj∈W, f

zj

f

zj

· 1

P1e^zj Q1e^−zj1 P₂e^zj Q₂e^−zj P1e^zj Q1e^−zj· f

zj

f

zj

1 f

zj

·R₁e^zj R₂e^−zj P1e^zj Q1e^−zj.

3.6 It follows from3.4–3.6andρ_ef 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

D

z:−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

will be a subnormal solution of1.7. Since we suppose that degP1 >degP2, we will discuss the following two cases.

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 e^β1zG₁e^zsee11, page 382, whereβ₁is a constant andG₁zis analytic on 0<|z|<∞. Thus,fzcan be written asfz e^β2zG₂e^z, whereβ₂ is a constant andG2zis analytic on 0<|z|<∞. It follows fromLemma 2.4that

fz e^βz

g₁e^z 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^−z

a /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,

fz−fzn·2πi na−→ ∞, 3.21

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 anda_kk0,1, . . . , mare constants witha_m/0.

LetndegP1degR1and set

P1e^{z n}

k0

pke^kz, R1e^{z n}

k0

rke^kz, 3.25

wherep_nr_n/0.

Now, we will discuss the following two cases.

Case A. We considera₀/0 in3.24. Letc₁be a constant defined by

c₁a₀p_nr_n, 3.26

and set

H1z c1zhz. 3.27

Sincehzis a subnormal solution of1.7, it follows from3.27thatH₁zsatisfies H₁

P₁e^z Q₁ e^−z

H₁

P₂e^z Q₂ e^−z

H₁2c₁hz c₁hz

P₁e^z Q₁ e^−z

. 3.28

We obtain from3.23–3.28that H₁

P₁e^z Q₁ e^−z

H₁

P₂e^z Q₂ e^−z

H₁r_ne^nzT₃e^z 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 P1e^z Q₁e^−zφ₁ P2e^z Q2e^−zφ1T3e^z T4e^−z−ⁿ⁻¹

k0rke^kz−R2e^−z. Set

S1e^z T3e^z−ⁿ⁻¹

k0

rke^kz, S2

e^−z T4

e^−z

−R2

e^−z

. 3.31

So degS₁≤n−1<degP₁, andφ₁zsatisfies φ₁

P₁e^z Q₁ e^−z

φ₁

P₂e^z Q₂ e^−z

φ₁ S₁e^z 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 e^βz

g₁e^z g₂ e^−z

, 3.33

whereβis a constant,g₁zandg₂zare polynomials inz. Hence3.23,3.24,3.27,3.30, and3.33yield

fz e^βz

−g1e^z−g₂ e^−z

c₁zg₃ 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

c₂sa_sp_n r_n, 3.35

wheres ∈ {0,1, . . . , m}is a number such thata_sis the first coefficienta₀, a₁, . . . , a_min3.24 which is not equal to zero. Set

H2z c2e^szhz. 3.36

Similar to the proof of CaseAofTheorem 1.1ii, we have fz e^βz

−g1e^z−g2

e^−z c2

e^szg3

e^−z

, 3.37

wherec₂/0 andβare constants,g₁z, g2zand,g₃zare polynomials inzwith degg₃≥1.

Setg4e^−z c2e^szg3e^−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 e^βz

g1e^z g2

e^−z

, 3.38

whereβis a constant,g₁zandg₂zare polynomials inz. By2.10and3.38, we obtain that

fz g0e^z e^βz

g1e^z 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 e^βz

g₁e^z g₂ e^−z

c₁zg₃ e^−z

c₂g₄ 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 f

e^2ze^−z1 f−

e^z2e^−z−1

f −e^z−32e^−2z. 3.41 This is an example ofTheorem 1.1i.

Example 3.2. Letfz z1e^−z e^−2z, thenfzsatisfies f

e^2z2e^ze^−z3 f

e^ze^−z1

f e^2z3e^z3−e^−ze^−3z. 3.42 This is an example ofTheorem 1.1iiwith degP₁>degP₂and degP₁degR₁.

Example 3.3. Letfz ze^−z e^ze^−z, thenfzsatisfies f

e^2ze^ze^−z f

e^ze^−z

fe^3z2e^2ze^z−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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