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On the Hyper-Order of Solutions of Some Second-Order Linear Differential Equations

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Zong-Xuan Chen
Zong-Xuan Chen
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Abstract

In this paper, we investigate the growth of solutions of the second-order linear homogeneous differential equations with entire coefficients of the same order, and obtain precise estimates of the hyper-order of their solutions.
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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 635604, 9pages
doi:10.1155/2011/635604
Research Article
On the Growth of Solutions of Some Second-Order
Linear Differential Equations
Feng Peng and Zong-Xuan Chen
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
Correspondence should be addressed to Zong-Xuan Chen, chzx@vip.sina.com
Received 10 December 2010; Accepted 9 February 2011
Academic Editor: Alberto Cabada
Copyright q2011 F. Peng and Z.-X. Chen. 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 investigate the growth of solutions of f PzfQzf0, where Pzand Qzare entire
functions. When Pze−zand QzA1zea1zA2zea2zsatisfy some conditions, we prove
that every nonzero solution of the above equation has infinite order and hyper-order 1, which
improve the previous results.
1. Introduction and Results
In this paper, we will assume that the reader is familiar with the fundamental results and the
standard notations of the Nevanlinna’s value distribution theory of meromorphic functions
e.g., see 1–3. In addition, we will use the notation σfto denote the order of growth of
meromorphic function fz,σ2fto denote the hyper-order of fzsee 3.σ2fis defined
to be
σ2flim
r→∞
log log Tr, f
log r.1.1
We consider the second-order linear differential equation
f PzfQzf0,1.2
where Pzand Qzare entire functions of finite order. It is well known that each solution
of 1.2is an entire function, and most solutions of 1.2have infinite order.
2 Journal of Inequalities and Applications
Thus, a natural question is what conditions on Pzand Qzwill guarantee that every
solution f/
≡0of 1.2has infinite order? Ozawa 4, Gundersen 5, Amemiya and Ozawa
6, and Langley 7have studied the problem with Pze−zand Qzis complex number
or polynomial. For the case that Pze−z,andQzis transcendental entire function,
Gundersen proved the following in 5, Theorem A.
Theorem A. If Qzis a transcendental entire function with order σQ/
1, then every solution
f/
≡0of equation
f e−zfQzf01.3
has infinite order.
Theorem Astates that when σQ1, 1.3may have finite-order solutions. We go
deep into the problem: what condition in Qzwhen σQ1 will guarantee every solution
f/
≡0of 1.3has infinite order? And more precise estimation for its rate of growth is a very
important aspect. Chen investigated the problem and obtain the following in 8, Theorem B
and Theorem C.
Theorem B. Let Ajz /
≡0j0,1be entire functions with σAj<1, and let a, b be complex
numbers such that ab /
0and acb c>1. then every solution f/
≡0of the equation
f A1eazfA0ebz f01.4
has infinite order.
Theorem C. Let a, b be nonzero complex numbers and a/
b, and let Qzbe a nonconstant
polynomial or Qzhzebz where hzis nonzero polynomial, then every solution f/
≡0of
the equation
f eazfQzf01.5
has infinite order and σ2f1.
For Theorems B and C, many authors, Wang and L¨
u9, Huang, Chen, and Li 10,
and Cheng and Kang 11have made some improvement. In this paper, we are concerned
with the more general problem, and obtain the following theorem that extend and improve
the previous results.
Theorem 1.1. Let Ajz/
≡0j1,2be entire functions with σAj<1,a
1,a
2be complex
numbers such that a1a2/
0, and let a1/
a2(suppose that |a1|≤|a2|.Ifarg a1/
πor a1<−1,
then every solution f/
≡0of the equation
f e−zfA1ea1zA2ea2zf01.6
has infinite order and σ2f1.
Journal of Inequalities and Applications 3
2. Remarks and Lemmas for the Proof of Theorem
Lemma 2.1 see 12.Let fbe a transcendental meromorphic function with σfσ<∞,H
{k1,j
1,k2,j
2,...,kq,j
q}be a finite set of distinct pairs of integers satisfying ki>j
i≥0i
1,2,...,q.Andletε>0be a given constant. Then,
ithere exists a set E⊂−π/2,3π/2with linear measure zero, such that, if ψ∈
−π/2,3π/2\E, then there is a constant R0R0ψ>1, such that for all zsatisfying
arg zψand |z|≥R0and for all k, j∈H, one has
fkz
fjz
≤|z|k−jσ−1ε,2.1
iithere exists a set E⊂1,∞with finite logarithmic measure, such that for all zsatisfying
|z|/∈E∪0,1and for all k, j∈H, we have
fkz
fjz
≤|z|k−jσ−1ε,2.2
iiithere exists a set E⊂0,∞with finite linear measure, such that for all zsatisfying |z|/∈E
and for all k, j∈H, we have
fkz
fjz
≤|z|k−jσε.2.3
Lemma 2.2 see 8.Suppose that Pzαiβzn··· α, β are real numbers, |α||β|/
0
is a polynomial with degree n≥1, that Az /
≡0is an entire function with σA<n.Setgz
AzePz,z reiθ,δP, θαcos nθ −βsin nθ. Then for any given ε>0, there exists a set
H1⊂0,2πthat has the linear measure zero, such that for any θ∈0,2π\H1∪H2,thereis
R>0, such that for |z|r>R, we have
iif δP, θ>0,then
exp{1−εδP, θrn}<
greiθ
<exp{1εδP, θrn},2.4
iiif δP, θ<0,then
exp{1εδP, θrn}<
greiθ
<exp{1−εδP, θrn},2.5
where H2{θ∈0,2π;δP, θ0}is a finite set.
4 Journal of Inequalities and Applications
Using Lemma 2.2, we can prove Lemma 2.3.
Lemma 2.3. Suppose that n≥1is a positive entire number. Let Pjzajnzn··· j1,2
be nonconstant polynomials, where ajq q1,2,...,nare complex numbers and a1na2n/
0.Set
zreiθ ,a
jn |ajn|eiθj,θ
j∈−π/2,3π/2,δPj,θ|ajn|cosθjnθ, then there is a set
H1⊂−π/2n,3π/2nthat has linear measure zero. If θ1/
θ2, then there exists a ray arg zθ,
θ∈−π/2n,π/2n\H1∪H2, such that
δP1,θ
>0,δ
P2,θ
<0,2.6
or
δP1,θ
<0,δ
P2,θ
>0,2.7
where H2{θ:θ∈−π/2n,3π/2n,δPj,θ0}is a finite set, which has linear measure zero.
Proof. According to the values of θ1and θ2, we divide our discussion into three cases.
Case 1 θ1∈−π/2,π/2.aIf θ2∈−π/2,π/2,letα1min{π/2−θ1,θ
1π/2},α
2
min{π/2−θ2,θ
2π/2}, Then there are three cases: iα1α2;iiα1<α
2;iiiα1>α
2.
iα1α2.Byθ1/
θ2, we know that θ1−θ2/
0.
Suppose that θ1>0, then take θ1/nπ/2−θ1t,tis any constant in 0,θ
1.
Since H1∪H2has linear measure zero, there exists t∈0,θ
1such that θ
1/nπ/2−θ1t∈0,π/2n\H1∪H2.Thusnθ π/2−θ1t∈0,π/2.By
θ1−θ2and θ1>0thatisθ1∈0,π/2, we have
θ1nθ π
2t∈π
2,π,θ
2nθ π
2−2θ1t∈−π
2,π
2.2.8
Therefore,
δP1,θ
|a1n|cosθ1nθ<0,δ
P2,θ
|a2n|cosθ2nθ>0.2.9
When θ1<0, then θ2>0, we can prove it by using similar argument action as in the
above proof.
iiα1<α
2, then θ1/
0. Suppose that θ1>0, then θ1>θ
2,0<θ
1−θ2<π.Let
θ0min{θ1,θ
1−θ2}, and take θ1/nπ/2−θ1t,and tis any constant in 0,θ
0.
Since H1∪H2has a linear measure zero, there exists t∈0,θ
0such that θ
1/nπ/2−θ1t∈0,π/2n\H1∪H2,
θ1nθ π
2t∈π
2,π,θ
2nθ π
2−θ1−θ2t∈−π
2,π
2.2.10
Therefore
δP1,θ
|a1n|cosθ1nθ<0,δ
P2,θ
|a2n|cosθ2nθ>0.2.11
Journal of Inequalities and Applications 5
Suppose that θ1<0, then θ1<θ
2,0<θ
2−θ1<π.Letθ0min{−θ1,θ
2−θ1}, and take
θ1/n−π/2−θ1−t,and tis any constant in 0,θ
0.
Since H1∪H2has linear measure zero, there exists t∈0,θ
0such that θ
1/n−π/2−θ1−t∈−π/2n, 0\H1∪H2,
θ1nθ −π
2−t∈−π, −π
2,θ
2nθ −π
2θ2−θ1−t∈−π
2,π
2.2.12
Therefore,
δP1,θ
|a1n|cosθ1nθ<0,δ
P2,θ
|a2n|cosθ2nθ>0.2.13
iiiα1>α
2, then θ2/
0. Using similar method as in proof of ii, we know that there
exists θ∈−π/2n,π/2n\H1∪H2such that δP1,θ>0,δP2,θ<0.
bWhen θ2∈π/2,3π/2, we can prove it by using the same argument action as in
a.
cWhen θ2∈{π/2,−π/2}, we just prove the case that θ2π/2when θ2−π/2,
we can prove it by using the same reasoning.
Let θ0min{π/2,π/2−θ1}, take θt/n, t is any constant in 0,θ
0.
Since H1∪H2has a linear measure zero, there exists t∈0,θ
0, such that θt/n ∈
0,π/2n\H1∪H2. Then
θ2nθ θ2t∈π
2,π.2.14
When θ1∈−π/2,0,t∈0,π/2,thus,−π/2<θ
1nθ θ1t<π/2.
When θ1∈0,π/2,t∈0,π/2−θ1,thus,0<θ
1nθ θ1t<θ
1π/2−θ1π/2.
Therefore
θ1nθ θ1t∈−π
2,π
2,
δP1,θ
|a1n|cosθ1nθ>0,δ
P2,θ
|a2n|cosθ2nθ<0.
2.15
Case 2. When θ1∈π/2,3π/2,orθ1∈{π/2,−π/2}and θ2/∈{π/2,−π/2},usingaproof
similar to Case 1, we can get the conclusion.
Case 3 θ1∈{π/2,−π/2}and θ2∈{π/2,−π/2}.Byθ1/
θ2, there are only two cases: θ1
π/2,θ
2−π/2; or θ1−π/2,θ
2π/2.
If θ1π/2,θ
2−π/2. Take θt/n, and tis any constant in 0,π/2.
Since H1∪H2has linear measure zero, there exists t∈0,π/2such that θt/n ∈
0,π/2n\H1∪H2. Using a proof similar to Case 1c, we can prove it.
When θ1−π/2,θ
2π/2, we can prove it by using the same reasoning
Remark 2.4. Using the similar reasoning of Lemma 2.3, we can obtain that, in Lemma 2.3,if
θ∈−π/2n, π/2n\H1∪H2is replaced by θ∈π/2n, 3π/2n\H1∪H2, then it has the
same result.
6 Journal of Inequalities and Applications
Lemma 2.5 see 8.Let A, B be entire functions with finite order. If fzis a solution of the
equation
f AfBf 02.16
then σ2f≤max{σA,σB}.
Lemma 2.6 see 12.Let fbe a transcendental meromorphic function, and let α>1be a given
constant, Then there exists a set E⊂1,∞with finite logarithmic measure and a constant B>0
that depends only on αand i, j (0≤i<j≤2), such that for all zsatisfying |z|r/∈0,1∪E,
fjz
fiz
≤BTαr, f
rlogαrlog Tαr, f j−i
.2.17
Remark 2.7. In Lemma 2.6, when α2,i0, we have
fjz
fz
≤BT2r, f log T2r, fj≤BT2r, f j1,j1,2.2.18
Lemma 2.8 see 13.Suppose that g:0,∞→Rand h:0,∞→Rare nondecreasing
functions, such that gr≤hr,r/∈E,whereEis a set with at most finite measure, then for any
constant α>1,thereexistsr0>0such that gr≤hαrfor all r>r
0.
3. Proof of Theorem 1.1
Suppose that f/
≡0is a solution of 1.6, then, fis an entire function.
First Step
We prove that σf∞. Suppose, to the contrary, that σfσ<∞.ByLemma 2.1,for
any given ε0<ε<|a2|−|a1|/|a2||a1|, there exists a set E1⊂−π/2,3π/2of linear
measure zero, such that if θ∈−π/2,3π/2\E1, then, there is a constant R0R0θ>1,
such that for all zsatisfying arg zθand |z|≥R0, we have
fz
fz
≤|z|2σ−1ε,
fz
fz
≤|z|σ−1ε.3.1
Let zreiθ ,a
1|a1|eiθ1,a
2|a2|eiθ2,θ
1,θ
2∈−π/2,3π/2.
Case 1 arg a1/
π, which is θ1/
π.iSuppose that θ1/
θ2. By Lemmas 2.2 and 2.3,forthe
above ε, there is a ray arg zθ, such that θ∈−π/2,π/2\E1∪H1∪H2where H1and
H2are defined as in Lemma 2.3,andE1∪H1∪H2is of the linear measure zero, and satisfying
δa1z, θ>0,δ
a2z, θ<0,3.2
Journal of Inequalities and Applications 7
or
δa1z, θ<0,δ
a2z, θ>0.3.3
When δa1z, θ>0,δa2z, θ<0, for sufficiently large r, we have
|A1ea1z|≥exp{1−εδa1z, θr},|A2ea2z|≤exp{1−εδa2z, θr}≤1.3.4
Hence
|A1ea1zA2ea2z|≥|A1ea1z|−|A2ea2z|≥exp{1−εδa1z, θr}−1.3.5
By 1.6,weobtain
fz
fz
e−z
fz
fz
≥|A1ea1zA2ea2z|.3.6
Since θ∈−π/2,π/2,weknowthatcosθ>0, then e−rcosθ<1. Substituting 3.1and 3.5
into 3.6,weget
r2σ−1εe−rcos θrσ−1ε≥exp{1−εδa1z, θr}−1,
2r2σ−1ε≥exp{1−εδa1z, θr}−1.
3.7
By δa1z, θ>0, we know that 3.7is a contradiction.
When δa1z, θ<0,δa2z, θ >0, using a proof similar to the above, we can also get a
contradiction.
iiSuppose that θ1θ2.ByLemma 2.2, for the above ε, there is a ray arg zθsuch
that θ∈−π/2,π/2\E1∪H1∪H2and δa1z, θ>0. Since |a1|≤|a2|,a
1/
a2,andθ1θ2,
then |a1|<|a2|,thusδa2z, θ >δa1z, θ>0. For sufficiently large r, we have
|A1ea1z|≤exp{1εδa1z, θr},|A2ea2z|≥exp{1−εδa2z, θr}.3.8
Hence,
|A1ea1zA2ea2z|≥|A2ea2z|−|A1ea1z|≥exp{1−εδa2z, θr}−exp{1εδa1z, θr}
≥M1exp{1εδa1z, θr},
3.9
where M1exp{1−εδa2z, θ−1εδa1z, θr}−1.
Since 0 <ε<|a2|−|a1|/|a2||a1|,weseethat1−εδa2z, θ−1εδa1z, θ >0,
then exp{1−εδa2z, θ−1εδa1z, θr}>1,M
1>0.
8 Journal of Inequalities and Applications
Since θ∈−π/2,π/2, we know that cos θ>0, then e−rcos θ<1. Substituting 3.1
and 3.9into 3.6,weobtain
r2σ−1εe−rcos θrσ−1ε≥M1exp{1εδa1z, θr},
2r2σ−1ε≥M1exp{1εδa1z, θr}.
3.10
Since δa1z, θ>0, we know that 3.10is a contradiction.
Case 2 a1<−1, which is θ1π.iSuppose that θ1/
θ2, then θ2/
π.ByLemma 2.2,forthe
above ε, there is a ray arg zθsuch that θ∈−π/2,π/2\E1∪H1∪H2and δa2z, θ>0.
Because cos θ>0,δa1z, θ|a1|cosθ1θ−|a1|cos θ<0. For sufficiently large r,we
have
|A1ea1z|≤exp{1−εδa1z, θr}≤1,|A2ea2z|≥exp{1−εδa2z, θr}.3.11
Hence
|A1ea1zA2ea2z|≥|A2ea2z|−|A1ea1z|≥exp{1−εδa2z, θr}−1.3.12
Using the same reasoning as in Case 1i, we can get a contradiction.
iiSuppose that θ1θ2π.ByLemma 2.2, for the above ε, there is a ray arg zθ
such that θ∈π/2,3π/2\E1∪H1∪H2, then cos θ<0,δa1z, θ −|a1|cos θ>0,δa2z, θ
−|a2|cos θ>0, Since |a1|≤|a2|,a
1/
a2and θ1θ2, then |a1|<|a2|.Thus,δa1z, θ<δa2z, θ,
for sufficiently large r,wegetthat3.8and 3.9hold.
Since a1<−1,cos θ<0, then δa1z, θ−|a1|cos θ>−cos θ>0.
Using the same reasoning as in Case 1ii, we can get a contradiction.
Concluding the above proof, we obtain σf∞.
Second Step
We prove that σ2f1.
By Lemma 2.5 and max{σe−z,σA1ea1zA2ea2z}1, then σ2f≤1.
By Lemma 2.6 and Remark 2.7, we know that there exists a set E2⊂1,∞with finite
logarithmic measure and a constant B>0, such that for all zsatisfying |z|r/∈0,1∪E2,
we get that 2.18holds.
For Cases 1and 2iin first step, we have proved that there is a ray argzθsatisfying
θ∈−π/2,π/2\E1∪H1∪H2,forsufficiently large r,wegetthat3.5or 3.9or 3.12
hold, that is,
|A1ea1zA2ea2z|≥exp{h1r},3.13
where h1>0 is a constant.
Since θ∈−π/2,π/2\E1∪H1∪H2, then cos θ>0,e
−rcos θ<1. By 2.18,3.6,
and 3.13,weobtain
exp{h1r}≤BT2r, f 3e−rcosθBT2r, f 2≤2BT2r, f3.3.14
Journal of Inequalities and Applications 9
By h1>0, 3.14and Lemma 2.8, we know that there exists r0, when r>r
0, we have σ2f≥1,
then σ2f1.
For Case 2iiin first step, we have proved that there is a ray argzθsatisfying
θ∈π/2,3π/2\E1∪H1∪H2,forsufficiently large r,weget3.9hold, and we also get
that cos θ<0,δa1z, θ>−cos θ>0.
By 2.18,3.6,and3.9,weobtain
M1exp{1εδa1z, θr}≤BT2r, f 3e−rcosθBT2r, f 2,
M1exp{1εδa1z, θr}≤2e−rcos θBT2r, f 3.
3.15
By δa1z, θ>−cos θ>0,M
1>0and3.15and Lemma 2.8, we know that there exists r0,
when r>r
0, we have σ2f≥1, then σ2f1.
Concluding the above proof, we obtain σ2f1.
Theorem 1.1 is thus proved.
Acknowledgment
This project was supported by the National Natural Science Foundation of China no.
10871076.
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... For the above question, some mathematicians investigated the second order linear differential equations and obtained many results (see REF. [2,3,5,6,10,15,16,20,24]). In 2002, Chen [3] considered the question: what condition on Q(z) when σ(Q) = 1 will guarantee every nontrivial solution of (1) has infinite order? ...
... In 2008, Wang and Laine [22] investigated the non-homogeneous equation related to (2) and obtained the following result. ...
... Proof: We rewrite the equation as [2], for any positive number (1)) (j = 0, 1, . . . , k). ...
Oscillation of solutions of some higher order linear differential equations
Article
  • Nov 2009
  • ELECTRON J QUAL THEO
In this paper, we deal with the order of growth and the hyper order of solutions of higher order linear differential equations where Bj (z) (j = 0, 1, ..., k - 1) and F are entire functions or polynomials. Some results are obtained which improve and extend previous results given by Z.-X. Chen, J. Wang, T.-B. Cao and C.-H. Li.
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... In [6], Chen has investigated the second order linear differential equation ...
... Theorem 1 (see [6]). Let P (z) = n i=0 a i z i and Q (z) = n i=0 b i z i be nonconstant polynomials where a i , b i (i = 0, 1, · · · , n) are complex numbers, a n b n = 0, let A 1 (z) , A 0 (z) ( ≡ 0) be entire functions. ...
Growth and oscillation related to a second-order linear differential equation
Article
Full-text available
  • May 2013
This paper is devoted to studying the growth and the oscillation of solutions of the second-order nonhomogeneous linear differential equation f '' +A 1 (z)e P(z) f ' +A 0 (z)e Q(z) f=F, where P(z),Q(z) are nonconstant polynomials such that deg P = deg Q = n and A j (z)(¬≡0) for j=0,1 and F(z) are entire functions with max{ρ(A j ):j=0,1}<n. We also investigate the relationship between small functions and differential polynomials g f (z)=d 2 f '' +d 1 f ' +d 0 f, where d 0 (z),d 1 (z),d 2 (z) are entire functions such that at least one of d 0 ,d 1 ,d 2 does not vanish identically with ρ(d j )<n for j=0,1,2 generated by solutions of the above equation.
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... with A(z) and B(z)( ≡ 0) are entire functions, the reader is referred to [1,7,8,11,14]. ...
Oscillation of meromorphic solutions to linear differential equations with coefficients of [p,q]-order
Article
Full-text available
  • Mar 2014
We study the relationship between "small functions" and the derivative of solutions to the higher order linear differential equation f(k)+Ak−1f(k−1)+⋯+A0f=0,(k≥2) f^{(k)}+A_{k-1}f^{(k-1)}+\dots+A_0f=0,\quad (k\geq 2) Here Aj(z)  (j=0,1,…,k−1)A_j(z)\; (j=0,1,\dots,k-1) are entire functions or meromorphic functions of [p,q]-order.
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On the Growth of Solutions of Higher Order Linear Homogeneous and Non-homogeneous Differential Equations
Article
Full-text available
  • Oct 2022
  • J NONLINEAR MATH PHY
In this paper, we discuss the growth of meromorphic solutions of some classes of homogeneous and nonhomogeneous differential equations. We will prove that if meromorphic functions F,Aj,DjF, A_j, D_j F , A j , D j and polynomials PjP_j P j with degree n≥1n\ge 1 n ≥ 1 (j=0,1,⋯ ,k−1)(j=0,1,\cdots ,k-1) ( j = 0 , 1 , ⋯ , k - 1 ) satisfy some conditions, then the equation f(k)+(Ak−1ePk−1+Dk−1)f(k−1)+⋯+(A1eP1+D1)f′+(A0eP0+D0)f=F(z)(k≥2)f^{(k)}+(A_{k-1}e^{P_{k-1}}+D_{k-1})f^{(k-1)}+\cdots +(A_1e^{P_1}+D_1)f'+(A_0e^{P_0}+D_0)f=F(z) (k\ge 2) f ( k ) + ( A k - 1 e P k - 1 + D k - 1 ) f ( k - 1 ) + ⋯ + ( A 1 e P 1 + D 1 ) f ′ + ( A 0 e P 0 + D 0 ) f = F ( z ) ( k ≥ 2 ) when F≡0F\equiv 0 F ≡ 0 , all solutions f≢0f\not \equiv 0 f ≢ 0 have infinite order and hyper order σ2(f)≥n\sigma _2(f)\ge n σ 2 ( f ) ≥ n . This is a continue work of [Gan and Sun in Adv. Math. 36 (1), 51–60 (2007)] and [Chen and Xu in Electron. J. Qual. Theory Differ. Equ. 1 : 1–13 (2009)]
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On hyper-order of solutions of higher order linear differential equations with meromorphic coefficients
Article
Full-text available
  • Apr 2016
  • ADV DIFFER EQU-NY
In this paper, we investigate the growth of meromorphic solutions of the differential equations f(k)+Ak−1(z)f(k−1)+⋯+A1(z)f′+A0(z)f=0f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=0 and f(k)+Ak−1(z)f(k−1)+⋯+A1(z)f′+A0(z)f=F(z),f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=F(z), where A0(z)≢0,A1(z),…,Ak−1(z)A_{0}(z)\not\equiv0, A_{1}(z), \ldots, A_{k-1}(z) and F(z)≢0F(z)\not \equiv0 are meromorphic functions. A precise estimation of the hyper-order of meromorphic solutions of the above equations is given provided that there exists one dominant coefficient, which improves and extends previous results given by Belaïdi, Chen, etc.
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MEROMORPHIC SOLUTIONS OF DIFFERENCE EQUATION $f(z+1)=R\circ f(z)
Article
  • Jan 2014
In this paper, we investigate the solutions of difference equation f(z + 1) = R o f (z) by utilizing Nevanlinna theory, where R(z) is a rational function. And we also research the quantity of zeroes, poles, fixed points, and Borel exceptional values of the solutions.
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On the Zeros of Solutions of Any Order of Derivative of Second Order Linear Differential Equations Taking Small Functions
Article
  • Apr 2011
  • ELECTRON J QUAL THEO
In this paper, we investigate the hyper-exponent of convergence of zeros of f (j)(z) -φ(z)(j ∈ N), where f is a solution of second or k(≥ 2) order linear differential equation, φ (z) 6≡ 0 is an entire function satisfying σ(φ) < σ(f) or σ2(φ) < σ2(f). We obtain some precise results which improve the previous results in [3, 5] and revise the previous results in [11, 13]. More importantly, these results also provide us a method to investigate the hyperexponent of convergence of zeros of f (j) (z) -φ (z)(j ∈ N).
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Zeros of solutions of certain higher order linear differential equations
Article
  • Jan 2010
We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation (1) f (k)+a k-1(z)f (k-1)+...+a 1(z)f′+D(z)f=0, where D(z)=Q 1(z)e P1(z)+Q 2(z)e P2(z)+Q 3(z)e P3(z), P 1(z),P 2(z),P 3(z) are polynomials of degree n≥1, Q 1(z),Q 2(z),Q 3(z),a j(z) (j=1,..., k-1) are entire functions of order less than n, and k≥2.
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Existence of entire solution of some certain type difference equation
Article
  • Jan 2013
  • HOUSTON J MATH
In this paper, we investigate the existence and growth of the solutions of some dierence equations by utilizing Nevanlinna theory and Wiman-Valiron theorem.
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Oscillation of Complex High Order Linear Differential Equations with Coefficients of Finite Iterated Order
Article
Full-text available
  • Dec 2009
  • ELECTRON J QUAL THEO
We investigate the growth of solutions of complex high order linear differential equations with entire or meromorphic coefficients of finite iterated order, and we obtain some results which improve and extend some previous results of Z. X. Chen [Acta Math. Sin., Engl. Ser. 18, No.1, 79–88 (2002; Zbl 1014.34078)] and L. Kinnunen [Southeast Asian Bull. Math. 22, No.4, 385–405 (1998; Zbl 0934.34076)].
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On the growth of solutions of f″ + gf′ + hf = 0
Article
  • Apr 1991
Suppose g and h are entire functions with the order of h less than the order of g. If the order of g does not exceed ½, it is shown that every (necessarily entire) nonconstant solution f of the differential equation f″ + gf′ + hf = 0 has infinite order. This result extends previous work of Ozawa and Gundersen.
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Finite order solutions of second order linear differential equations
Article
  • Jan 1988
We consider the differential equation /" + A(z)f' B(z)f = 0 where A(z) and B(z) are entire functions. We will find conditions on A(z) and B(z) (Formula present) which will guarantee that every solution 0 of the equation will have infinite order. We will also find conditions on A(z) and B[z) which will guarantee that any finite order solution 0 of the equation will not have zero as a Borel exceptional value. W e will also show that if A(z) and B(z) satisfy certain growth conditions, then any finite order solution of the equation will satisfy certain other growth conditions. Related results are also proven. Several examples are given to complement the theory.
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The growth of solutions of f '' +e -z f ' +Q(z)f=0 where the order (Q)=1
Article
  • Mar 2002
This paper investigates the growth of solutions of the equation f '' +e-zf ' +Q(z)f=0 where the order (Q)=1. When Q(z)=h(z)e bz h(z) is a nonzero polynomial, b≠-1 is a complex constant, every solution of the above equation has infinite order and the hyper-order 1. We improve results due to M. Frei, M. Ozawa, G. Gundersen and J. K. Langley.
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On the zeros and hyper-order of meromorphic solutions of linear differential equations
Article
  • Jan 1999
In this paper we investigate the exponent of convergence of the zeros and the hyper-order of meromorphic solutions of higher-order homogeneous linear differential equations with transcendental coefficients.
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