R E S E A R C H
Open Access
On the growth of solutions of a class of
second-order complex differential equations
Cai Feng Yi
1*, Xu-Qiang Liu
1and Hong Yan Xu
2*Correspondence:
1Department of Mathematics and
Informatics, Jiangxi Normal University, Nanchang, Jiangxi 330022, P.R. China
Full list of author information is available at the end of the article
Abstract
In this paper, we consider the differential equationf+h(z)eP(z)f+Q(z)f= 0, where
h(z) andQ(z)≡0 are meromorphic functions,P(z) is a non-constant polynomial. Assume thatQ(z) has an infinite deficient value and finitely many Borel directions. We give some conditions onP(z) which guarantee that every solutionf≡0 of the equation has infinite order.
MSC: 34AD20; 30D35
Keywords: complex differential equation; meromorphic function; Borel direction; deficient value; hyper-order
1 Introduction and main results
In this paper, we shall involve the deficient value and theBoreldirection in investigating the growth of solutions of the second-order linear differential equation
f+h(z)eP(z)f+Q(z)f= , ()
whereh(z) andQ(z)≡ are meromorphic functions,P(z) is a non-constant polynomial. We assume that the reader is familiar with the Nevanlinna theory of meromorphic func-tions and the basic nofunc-tions such asN(r,f),m(r,f),T(r,f) andδ(r,f). For the details, see [] or [].
The orderσand the hyper-orderσare defined as follows:
σ(f) =lim sup r−→∞
log+T(r,f)
logr , σ(f) =lim supr−→∞
log+log+T(r,f)
logr .
It is well known that ifA(z) =h(z)eP(z) andB(z) =Q(z) are transcendental entire
func-tions in equation () andf,fare two linearly independent solutions of equation (), then
at least one off,fmust have infinite order. Hence, ‘most’ solutions of equation () will
have infinite order. On the other hand, there are some equations of the form () that pos-sess a solutionf ≡ which has finite order; for example,f(z) =ez satisfies the equation
f+e–zf– (e–z+ )f = . Thus the main problem is what condition onA(z) andB(z) can guarantee that every solutionf≡ of equation () has infinite order? There has been much work on this subject. For example, it follows from the work by Gunderson [], Hellerstein
() has infinite order. Furthermore, ifAis an entire function with finite order having a fi-nite deficient value andB(z) is a transcendental entire function withμ(B) <, then every solutionf ≡ of equation () has infinite order []. More results can be found in [–].
However, it seems that there is little work done on equation () whose coefficient func-tions are meromorphic funcfunc-tions. Recently, Wuet al.discussed the problem correlating with this in []. Now we still consider equation () with transcendental meromorphic co-efficients and discuss the growth of its meromorphic solutions. We shall also involve the deficient value and the Borel direction in the studies of the oscillation of the second-order complex differential equation. We hope that the relations between the orders of coefficient functions will not be restricted. In general, it would not hold that every solutionf ≡ of equation () has infinite order; for example,f(z) =ezz satisfies
f+ e z
z–zf
–ez– z+
z f =
andσ(f) = <∞.
To state our theorem, we give some remarks first. LetP(z) = (α+iβ)zn+· · · (α,β∈R) be a non-constant polynomial. Denoteδ(P,θ) =αcosnθ–βsinnθ, letdegPbe the degree ofP(z),(θ,ε,r) ={z:θ–ε<argz<θ+ε,|z|<r}. In the following, we give the definition of theBoreldirection of a meromorphic functionf(z).
Definition .[] Letf(z) be a meromorphic function in the complex plane withσ(f) =σ
( <σ≤ ∞). A rayargz=θ(≤θ< π) starting from the origin is called aBoreldirection of orderσoff(z) if the following equality:
lim sup r−→∞
logn((θ,ε,r),f=a)
logr =σ
holds for any real numberε> and every complex numbera∈C∪ {∞}with at most two exceptions.
The main results in this article are stated as follows.
Theorem . Let P(z)be a non-constant polynomial withdegP=n,let h(z)be a mero-morphic function withσ(h) <n.Suppose that Q(z)is a finite-order meromorphic function having an infinite deficient value,Q(z)has only finitely many Borel directions:Bj:argz=θj (j= , , . . . ,q).Denote thatj={z:θj<argz<θj+},j= , , . . . ,q.Suppose that there exists ϕj(θj<ϕj<θj+)such thatδ(P,ϕj) < for each angular domainj.Then every meromorphic
solution f ≡of equation()has infinite order andσ(f)≥σ(Q).
Remark . We apply the theorem to some particular equations. For example, when
Q(z) =g(z)ebz, whereg(z) is a non-constant polynomial andb= –. Chen proved [] that every meromorphic solutionf ≡ of the equation
f+e–zf+Q(z)f = () has infinite order withσ(f) = . Except the case ofargb= ,π, by the theorem, we can get
From the structure ofE={ϕ:δ(P,ϕ) < }andE={ϕ:δ(P,ϕ) > }in [, π), we can
easily get the following conclusion.
Corollary . Let P(z)be a non-constant polynomial withdegP=n,let h(z)be a mero-morphic function withσ(h) <n.Let Q(z)be a transcendental meromorphic function with finite order.If Q(z)has a deficient value∞and has only q Borel directions Bj:argz=θj (j= , , . . . ,q)that satisfyθ<θ<· · ·<θq<θq+(θq+=θ+ π)and
ω=min
≤j≤p{θj+–θj}>
π
n, ()
then every meromorphic solution f ≡of equation()has infinite order andσ(f)≥σ(Q).
By using the corollary, we see that ifσ(h) <n<degP, then every meromorphic solution
f ≡ of the equation
f+h(z)eP(z)f+eznf=
has infinite order withσ(f)≥n.
2 Some lemmas
To prove our theorem, we need the following lemmas.
Lemma .[] Let(f,)denote a pair that consists of a transcendental meromorphic function f(z)and a finite set
=(k,j), (k,j), . . . , (kq,jq)
of distinct pairs of integers that satisfy ki>ji≥for i= , , . . . ,q.Letα> andε> be
given real constants.Then the following three statements hold.
(i) There exists a setE⊂[, π)that has linear measure zero,and there exists a
constantc> that depends only onαandsuch that ifϕ∈[, π) –E,then there
is a constantR=R(ϕ) > such that for allzsatisfyingargz=ϕand|z|=r≥R,
and for all(k,j)∈,we have
ff((kj))((zz))
≤c
T(αr,f)
r log
α
rlogT(αr,f)
k–j
. ()
In particular,iff(z)has finite orderσ(f),then()is replaced by().
ff((kj))((zz))
≤ |z|(k–j)(σ(f)–+ε). ()
(ii) There exists a setE⊂(,∞)that has finite logarithmic measure,and there exists a
constantc> that depends only onαandsuch that for allzsatisfying
|z|=r∈/E∪[, ]and for all(k,j)∈,the inequality()holds.
(iii) There exists a setE⊂[,∞)that has finite linear measure,and there exists a
constantc> that depends only onαandsuch that for allzsatisfying|z|=r∈/E
and for all(k,j)∈,we have
ff((kj))((zz))
≤cT(αr,f)rεlogT(αr,f)k–j. ()
In particular,iff(z)has finite orderσ(f),then()is replaced by()
ff((kj))((zz))
≤ |z|(k–j)(σ(f)+ε). ()
Lemma .[] Suppose that g(z) =h(z)eP(z),where P(z)is a non-constant polynomial
withdegP=n,and h(z)is a meromorphic function withσ(h) <n.There exists a set E⊂
[, π)that has linear measure zero such that for allϕ∈[, π)\E,we have (i) Ifδ(P,ϕ) < ,then there is a constantR=R(ϕ) > such that the inequality
greiϕ<exp
δ(P,ϕ)r n
()
holds forr>R.
(ii) Ifδ(P,ϕ) > ,then there is a constantR=R(ϕ) > such that the inequality
greiϕ>exp
δ(P,ϕ)r n
()
holds forr>R.
Lemma .[] Let f(z)be a transcendental meromorphic function with finite order σ,
then there exists a functionλ(r)with the following properties:
(i) λ(r)is a non-negative and continuous function forr≥withlimr−→∞λ(r) =σ. (ii) λ(r)is a differentiable function for allrin(,∞)with at most countable exceptions
andlimr−→∞λ(r)logr= .
(iii) The inequalityrλ(r)≥T(r,f)holds for all sufficiently larger,and there exists a
sequencernwithrn→ ∞satisfyingrnλ(rn)=T(rn,f).
We shall call the functionλ(r)the proximate order of f(z),and the function U(r) =rλ(r)
the type function of f(z).
Lemma .[] Let f(z)be a transcendental meromorphic function with orderσ( <σ<
∞).B:argz=ϕand B:argz=ϕ(≤ϕ<ϕ≤π+ϕ)are two half rays starting from
the origin,and f has no Borel direction in the angular domainϕ<argz<ϕ.Suppose that
there exists a sequence rnwith rn→ ∞(n→ ∞)and a complex number a(a∈C∪ ∞)
such that the following inequality:
log| f(rneiϕ)–a|>r
σ–ε
n , a=∞,
log|f(rneiϕ)|>rnσ–ε, a=∞,
()
holds for any given constantε> and all sufficiently large n in some raysargz=ϕ,where
anyϕ∈En satisfies the inequality().If there exists a constant K> (not dependent on
(ii) For any positive numberη> ,the inequality
log|f(z)–a|>rnσ–η, a=∞,
log|f(z)|>rnσ–η, a=∞,
()
holds for sufficiently largenandz∈Ln.
Lemma . Let f(z)be a transcendental meromorphic function with orderσ having an
infinite deficient value.If f(z)has q Borel directions,Bj:argz=θj(j= , , . . . ,q),and these
holds for all sufficiently large n,where En,j= For everyrn, by the Boutroux-Cartan theorem [], we have
n(rn,f=∞)
ν=
|z–bν|> (hrn)n(rn,f=∞), ()
≤m(rn,f) +n(rn,f=∞)logrn–log
According to the properties ofU(r), we have
T(rn,f) < U(rn) < σ+U(rn). ()
From ()-(), we get
measEn≥
δπ
Kσ+.
Since the whole complex plane is divided intoqangular domains and there is noBorel
direction in them, the circle|z|=Rnis also divided intoqarcs:Anj:{Rneiϕ:θj<ϕ<θj+}
Obviously, we have
holds for all sufficiently largen.
We chooseK=qKδπσ+,K=ξ. By Lemma ., for all sufficiently largen, there exists a
The proof of Lemma . is completed.
3 Proof of Theorem 1.1
Proof Suppose thatf ≡ is a meromorphic solution of equation () withσ(f) <∞. We shall seek for a contradiction. From equation (), we have the following inequality:
Q(z)≤f of measure zero andR> such that the following inequality:
Denote thatj={z:θj<argz<θj+},j= , , . . . ,q. Applying Lemma . toQ(z), then for
any given constantsη> andξ> , there exists an angular domainjand a sequencerm withrm→ ∞(m→ ∞) such that () holds for all sufficiently largem.
On the other hand, since there existsϕjinjsuch thatδ(P,ϕj) < by the supposition of the theorem, we can get an interval [θ,θ]⊂j such that () holds for allz=reiϕ satisfyingr>Randϕ∈[θ,θ]\E. Now, letξ=
θ–θ
. For each sufficiently largem, we
can chooseϕm∈[θ,θ]\(E∪E) such that (), () and the inequality
logQ(zm)>rσm(Q)–η ()
hold forzm=rmeiϕn. Letη=σ(Q). Hence, from ()-(), we get
exprηm≤rmσ(f)
+exp
δ(P,ϕm)r n m
. ()
Obviously, whenmis sufficiently large, this is a contradiction. Next, we will proveσ(f)≥σ(Q).
By using Lemma ., there exist a setE⊂[, π) of measure zero and two constants
B> andR> such that for allz satisfying|z|=r>R andargz∈/ E, the following
inequality holds:
ff(j()z(z))≤BT(r,f). ()
Hence, for each sufficiently largem, we can chooseϕm ∈[θ,θ]\(E∪E) such that (),
() and () hold forzm=rmeiϕ
m. From (), (), () and (), we get
exprσ(Q)–η
m ≤BT(r,f)
+exp
δ
P,ϕmrmm
. ()
Thus
lim sup m→+∞
log+log+T(r,f)
logr ≥σ(Q) –η.
Asηcan be arbitrary small, we haveσ(f)≥σ(Q).
The proof of the theorem is completed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
CFY and XQL completed the main part of this article, CFY, XQL and HYX corrected the main theorems. All authors read and approved the final manuscript.
Author details
1Department of Mathematics and Informatics, Jiangxi Normal University, Nanchang, Jiangxi 330022, P.R. China.
2Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China.
Acknowledgements
The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (11171170, 61202313), the Natural Science Foundation of Jiang-Xi Province in China (Grant No. 2010GQS0119, No. 20132BAB211001 and No. 20122BAB201016).
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Cite this article as:Yi et al.:On the growth of solutions of a class of second-order complex differential equations.