R E S E A R C H
Open Access
Complex oscillation of a second-order linear
differential equation with entire coefficients
of
[
p
,
q
] –
ϕ
order
Xia Shen
1, Jin Tu
2and Hong Yan Xu
3**Correspondence:
3Department of Informatics and
Engineering, Jingdezhen Ceramic Institute, Jingdezhen, 333403, China Full list of author information is available at the end of the article
Abstract
In this paper, the authors investigate the interaction between the growth, zeros of solutions with the coefficients of second-order linear differential equations in terms of [p,q] –
ϕ
order and obtain some results in general form.MSC: 30D35; 34A20
Keywords: linear differential equations; [p,q] –
ϕ
order; [p,q] –ϕ
exponent of convergence of zero sequence1 Introduction and notations
In this paper, we shall assume that readers are familiar with the standard notations of Nevanlinna value distribution theory (see [–]). The theory of complex linear equations has been developed since s. Many authors have investigated the second-order linear differential equation
f+A(z)f= , (.)
whereA(z) is an entire function or a meromorphic function of finite order or finite iterated order, and have obtained many results about the interaction between the solutions and the coefficient of (.) (see [–]). What about the case whenA(z) is an entire function of [p,q ]-order or more general growth? In the following, we will introduce some notations about [p,q]-order, wherepandqare two positive integers and satisfyp≥q≥ throughout this paper (see [–]). Firstly, forr∈[, +∞), we defineexpr=erandexpi+r=exp(expir),
i∈N, and for all sufficiently larger, we definelogr=lograndlogi+r=log(logir),i∈N. Especially, we haveexpr=r=lograndexp–r=logr. Secondly, we denote the linear measure and the logarithmic measure of a setE⊂(, +∞) bymE=EdtandmlE=
E dt
t .
Definition .([]) Iff(z) is a meromorphic function, the [p,q]-order off(z) is defined by
σ[p,q](f) = lim r→∞
logpT(r,f)
logqr . (.)
Especially, iff(z) is an entire function, then the [p,q]-order off(z) is defined by (see [, , , ])
σ[p,q](f) = lim r→∞
logpT(r,f)
logqr =rlim→∞
logp+M(r,f)
logqr . (.)
Remark . We useσ[,](f) =σ(f) andσ[p,](f) =σp(f) to denote the order and the iterated order of a functionf(z).
Definition .([, ]) The growth index (or the finiteness degree) of the iterated order
of a meromorphic functionf(z) is defined by
i(f) =
⎧ ⎪ ⎨ ⎪ ⎩
iff is rational,
min{n∈N:σn(f) <∞} iff is transcendental andσn(f) <∞for somen∈N,
∞ if withσn(f) =∞for alln∈N.
Remark . By Definition ., we can similarly give the definition of the growth index
of the iterated exponent of convergence of the zero-sequence of a meromorphic function
f(z) byiλ(f, ).
Definition . ([, ]) The [p,q] exponent of convergence of the (distinct)
zero-sequence of a meromorphic functionf(z) is respectively defined by
λ[p,q](f) = lim r→∞
logpn(r,f)
logqr =rlim→∞
logpN(r,f)
logqr , (.)
λ[p,q](f) = lim r→∞
logpn(r,f)
logqr =rlim→∞
logpN(r,f)
logqr . (.)
Definition .([]) The [p,q] exponent of convergence of the (distinct) pole-sequence of a meromorphic functionf(z) is respectively defined by
λ[p,q]
f
= lim
r→∞
logpn(r,f)
logqr , (.)
λ[p,q]
f
= lim
r→∞
logpn(r,f)
logqr . (.)
Remark . We useλ[,](f) =λ(f),λ[p,](f) =λp(f) andλ[,](f) =λ(f),λ[p,](f) =λp(f) to denote the (iterated) exponent of convergence of the zero-sequence and pole-sequence of a meromorphic functionf(z).
Recently, some authors have investigated the exponent of convergence of the zero-sequence and pole-zero-sequence of the solutions of second-order linear differential equations (see [–]) and have obtained the following results.
Theorem A([]) Let A be a transcendental meromorphic function of orderσ(A),where
(.),we have
σ(A)≤max λ(f),λ
f
.
Theorem B([]) Let A(z)be an entire function with i(A) =p∈N+.Let f,fbe two linearly
independent solutions of(.)and denote F=ff.Then iλ(F, )≤p+ and
λp+(F, ) =σp+(F) =max
λp+(f, ),λp+(f, )
≤σp(A).
If iλ(F, )≤p,then iλ(f, ) =p+ holds for all solutions of type f =cf+cf,where cc= .
Theorem C([]) Let A(z)be an entire function with <i(A) =p<∞,let f be any non-trivial solution of(.),and assumeλp(A, ) <σp(A)= .Thenλp+(f, )≤σp(A)≤λp(f, ).
Theorem D([]) Let A(z)be an entire function with i(A) =p andσp(A) =σ<∞.Let f
and fbe two linearly independent solutions of(.)such thatmax{λp(f, ),λp(f, )}<σ.
Let(z) ≡be any entire function for which either i() <p or i() =p andσp() <σ.
Then any two linearly independent solutions g and g of the differential equation y+ (A(z) +(z))y= satisfymax{λp(g),λp(g)} ≥σ.
Theorem E([]) Let A be a meromorphic function with i(A) =p∈N+,and assume that
λp(A) <σp(A).Then,if f is a nonzero meromorphic solution of(.),we have
σp(A)≤max λp(f),λp
f
.
In the special case where eitherδ(∞,f) > or the poles of f are of uniformly bounded mul-tiplicities,we can conclude that
max λp+(f),λp+
f
≤σp(f)≤ λp(f),λp
f
.
In [], Chyzhykov and his co-authors introduced the definition ofϕ-order off(z), where
f(z) is a meromorphic function in the unit disc and used it to investigate the interaction between the analytic coefficients and solutions of
f(k)+Ak–(z)f(k–)+· · ·+A(z)f =
in the unit disc, where the definition ofϕ-order off(z) is given as follows.
Definition .([]) Letϕ: [, )→(, +∞) be a non-decreasing unbounded function,
theϕ-order of a meromorphic functionf(z) in the unit disc is defined by
σ(f,ϕ) = lim r→–
log+T(r,f)
logϕ(r) . (.)
Definition . Letϕ: [, +∞)→(, +∞) be a non-decreasing unbounded function, the [p,q] –ϕorder and [p,q] –ϕlower order of a meromorphic functionf(z) are respectively defined by
σ[p,q](f,ϕ) = lim r→∞
logpT(r,f)
logqϕ(r) , (.)
μ[p,q](f,ϕ) = lim r→∞
logpT(r,f)
logqϕ(r) . (.)
Similar to Definition ., we can also define the [p,q] –ϕ exponent of convergence of the (distinct) zero-sequence of a meromorphic functionf(z).
Definition . The [p,q] –ϕexponent of convergence of the (distinct) zero-sequence of a meromorphic functionf(z) is respectively defined by
λ[p,q](f,ϕ) = lim r→∞
logpn(r,f)
logqϕ(r) , (.)
λ[p,q](f,ϕ) = lim r→∞
logpn(r, f)
logqϕ(r) . (.)
Proposition . If f(z), f(z) are meromorphic functions satisfying σ[p,q](f,ϕ) = a, σ[p,q](f,ϕ) =b,then
(i) σ[p,q](f+f,ϕ)≤max{a,b},σ[p,q](f·f,ϕ)≤max{a,b};
(ii) Ifa=b,σ[p,q](f+f,ϕ) =max{a,b},σ[p,q](f·f,ϕ) =max{a,b}.
In this paper, we add two conditions onϕ(r) as follows:ϕ(r) : [, +∞)→(, +∞) is a non-decreasing unbounded function and satisfies (i)limr→∞loglogqp+ϕ(rr)= , (ii)limr→∞
logqϕ(αr)
logqϕ(r) = for someα> . Throughout this paper, we assume thatϕ(r) always satisfies the above two conditions without special instruction.
Proposition . Letϕ(r)satisfy the above two conditions(i)-(ii).
(i) Iff(z)is an entire function,then
σ[p,q](f,ϕ) = lim r→∞
logpT(r,f)
logqϕ(r) =rlim→∞
logp+M(r,f)
logqϕ(r) ,
μ[p,q](f,ϕ) = lim r→∞
logpT(r,f)
logqϕ(r) =rlim→∞
logp+M(r,f)
logqϕ(r) .
(ii) Iff(z)is a meromorphic function,then
λ[p,q](f,ϕ) = lim r→∞
logpn(r,f)
logqϕ(r) =rlim→∞
logpN(r,f)
logqϕ(r) ,
λ[p,q](f,ϕ) = lim r→∞
logpn(r,f)
logqϕ(r) =rlim→∞
logpN(r,f)
Proof (i) By the inequalityT(r,f)≤log+M(r,f)≤RR+–rrT(R,f) ( <r<R), setR=αr(α> ), to investigate the growth, zeros of the solutions of equation (.).
Theorem . Let A(z) be an entire function satisfying λ[p,q](A,ϕ) <σ[p,q](A,ϕ). Then λ[p+,q](f,ϕ)≤σ[p,q](A,ϕ)≤λ[p,q](f,ϕ)holds for all non-trivial solutions of(.).
Theorem . Let A(z)be an entire function satisfyingσ[p,q](A,ϕ) =σ > ,let f and f
be two linearly independent solutions of(.)such thatmax{λ[p,q](f,ϕ),λ[p,q](f,ϕ)}<σ.
Let(z) ≡be any entire function satisfyingσ[p,q](,ϕ) <σ.Then any two linearly
in-dependent solutions g and g of the differential equation f+ (A(z) +(z))f = satisfy max{λ[p,q](g,ϕ),λ[p,q](g,ϕ)} ≥σ.
3 Some lemmas
Lemma .([–]) Let f(z)be a transcendental entire function,and let z be a point with |z|=r at which|f(z)|=M(r,f).Then,for all|z|outside a set E of r of finite logarithmic
measure,we have
f(j)(z)
f(z) =
vf(r)
z j
+o() (j∈N), (.)
where vf(r)is the central index of f(z).
Lemma .([, , ]) Let g: [, +∞)−→Rand h: [, +∞)−→Rbe monotone
non-decreasing functions such that g(r)≤h(r)outside of an exceptional set E of finite linear
measure or finite logarithmic measure.Then,for any d> ,there exists r> such that
g(r)≤h(dr)for all r>r.
Lemma .([, ]) Let f(z) =∞n=anznbe an entire function,μ(r)be the maximum
term,i.e.,μ(r) =max{|an|rn;n= , , . . .},and let vf(r)be the central index of f.
(i) If|a| = ,then
logμ(r) =log|a|+
r
vf(t)
t dt. (.)
(ii) Forr<R,we have
M(r,f) <μ(r) vf(R) +
R R–r
. (.)
Lemma . Let f(z)be an entire function satisfyingσ[p,q](f,ϕ) =σandμ[p,q](f,ϕ) =μ,
and let vf(r)be the central index of f,then
lim r→∞
logpvf(r)
logqϕ(r) =σ, rlim→∞
logpvf(r)
logqϕ(r) =μ.
Proof Letf(z) =∞n=anzn. Without loss of generality, we can assume that|a| = . From (.), for any <α<α, we have
logμ(αr) =log|a|+
αr
vf(t)
t dt≥log|a|+ αr
r
vf(t)
t dt≥log|a|+vf(r)logα.
By the Cauchy inequality, it is easy to seeμ(αr)≤M(αr,f), hence
By Definition . and (.), we have
By the lemma of logarithmic derivative, we have
Lemma . Let f(z)be an entire function of[p,q] –ϕorder,and f(z)can be represented by the form
f(z) =U(z)eV(z),
where U(z)and V(z)are entire functions of[p,q] –ϕorder such that
λ[p,q](f,ϕ) =λ[p,q](U,ϕ) =σ[p,q](U,ϕ),
σ[p,q](f,ϕ) =max
σ[p,q](U,ϕ),σ[p,q]
eV,ϕ.
4 Proofs of Theorems 2.1-2.4
Proof of Theorem. Setσ[p,q](A,ϕ) =σ> . First, we prove that every solution of (.) satisfies σ[p+,q](f,ϕ)≤σ. If f(z) is a polynomial solution of (.), it is easy to know thatσ[p+,q](f,ϕ) = ≤σholds. Iff(z) is a transcendental solution of (.), by (.) and Lemma ., there exists a setE⊂(, +∞) having finite logarithmic measure such that for allzsatisfying|z|=r∈/[, ]∪Eand|f(z)|=M(r,f), we have
vf(r)
r
+o()≤expp+ σ+ε
logqϕ(r)
.
And hence, we have
vf(r)≤rexpp+
(σ+ε)logqϕ(r)
(r∈/E). (.)
By (.) and Lemma ., there exists someα( <α<α) such that for allr≥r, we have
vf(r)≤αrexpp+
(σ+ε)logqϕ(αr)
. (.)
By Lemma ., (.) and the two conditions onϕ(r), we have
σ[p+,q](f,ϕ) = lim r→∞
logp+vf(r)
logqϕ(r) ≤σ. (.)
On the other hand, by (.), we have
m(r,A) =m
r, –f
f
=OlogrT(r,f). (.)
By (.), we have σ[p,q](A,ϕ) ≤ σ[p+,q](f,ϕ). Therefore, we have that σ[p+,q](f,ϕ) =
σ[p,q](A,ϕ) holds for all non-trivial solutions of (.).
Proof of Theorem . Setσ[p,q](A,ϕ) =σ> , by Theorem ., we haveσ[p+,q](f,ϕ) = σ[p+,q](f,ϕ) =σ[p,q](A,ϕ) =σ. Hence, we have
λ[p+,q](F,ϕ)≤σ[p+,q](F,ϕ)≤max
σ[p+,q](f,ϕ),σ[p+,q](f,ϕ)
By Lemma . and (.), we have diction. Therefore, the first assertion is proved.
By Definition . and (.), we haveσ[p+,q](f,ϕ)≤β<σ[p,q](A,ϕ), this is a contradiction the functionA(z) implies that
By Definition . and (.), we haveσ[p,q](F,ϕ)≤σ. On the other hand, by
A=
F F
– F
F –
F, (.)
we haveσ[p,q](A,ϕ) =σ≤σ[p,q](F,ϕ), henceσ[p,q](F,ϕ) =σ. The same reasoning is valid for the functionF, we have
(A+) =
F F
– F
F
–
F
, (.)
andσ[p,q](F,ϕ) =σ. Sinceλ[p,q](F,ϕ) <σ andλ[p,q](F,ϕ) <σ, by Lemma ., we may write
F=QeP, F=ReS, (.)
whereP,Q,R,Sare entire functions satisfyingσ[p,q](Q,ϕ) =λ[p,q](F,ϕ) <σ,σ[p,q](R,ϕ) = λ[p,q](F,ϕ) <σandσ[p,q](eP,ϕ) =σ[p,q](eS,ϕ) =σ. Substituting (.) into (.) and (.), we have
A= –
QeP+G(z), (.)
(A+π) = –
ReS +G(z), (.)
where G(z) andG(z) are meromorphic functions satisfyingσ[p,q](Gj,ϕ) <σ (j= , ). Equation (.) subtracting (.), we have
ReS –
QeP=G(z), (.)
whereG(z) is a meromorphic function satisfyingσ[p,q](G,ϕ) <σ. From (.), we have
e–S+He–P=H, (.)
whereH(z) andH(z) are meromorphic functions satisfyingσ[p,q](Hj,ϕ) <σ(j= , ), and
H= –R
Q. Deriving (.), we have
–Se–S+H– PH
e–P=H, (.)
where H(z) is a meromorphic function satisfyingσ[p,q](H,ϕ) <σ. Eliminatinge–S by (.) and (.), we have
H– P–SH
e–P=H, (.)
whereH(z) is a meromorphic function satisfyingσ[p,q](H,ϕ) <σ. Sinceσ[p,q](eP,ϕ) =σ, therefore by (.), we haveH– (P–S)H≡, thus we haveH=ce(P–S),c= . Hence
F
F
=Q
Re
(P–S)= –
From (.), (.) and (.), we have
By Lemma ., we obtain
T
From (.) and (.), we have≡, this is a contradiction. Therefore, we obtain the
conclusion of Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
XS, JT and HYX completed the main part of this article, JT and HYX corrected the main theorems. All authors read and approved the final manuscript.
Author details
1College of Science, Jiujiang University, JiuJiang, 332005, China.2College of Mathematics and Information Science,
Jiangxi Normal University, Nanchang, 330022, China. 3Department of Informatics and Engineering, Jingdezhen Ceramic
Institute, Jingdezhen, 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 (11301233), the Natural Science Foundation of Jiang-Xi Province in China (20132BAB211001, 20132BAB211002), and the Foundation of Education Department of Jiangxi (GJJ14272, GJJ14644) of China.
Received: 22 February 2014 Accepted: 22 May 2014 Published:23 Jul 2014
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