R E S E A R C H
Open Access
On the hyper-order of solutions of two
class of complex linear differential equations
Wen Ping Huang
1, Jing Lun Zhou
1, Jin Tu
2*and Ju Hong Ning
2*Correspondence:
2College of Mathematics and
Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
Full list of author information is available at the end of the article
Abstract
We investigate the hyper-order of solutions of two class of complex linear differential equations. We investigate the growth of solutions of higher order and certain second order linear differential equations, and we obtain some results which improve and extend some previous results in complex oscillations.
MSC: 30D35; 34M10
Keywords: linear differential equation; entire function; hyper-order; maximum modulus
1 Introduction and results
We shall assume that reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions (e.g.see [, ]). In this paper, we useρ(f),τ(f) to denote the order and type of an entire function
f(z), useλ(f) (λ(f)) to denote the exponent of convergence of zeros (distinct zeros) off(z), and useρ(f) to denote the hyper-order off(z) (see []), which is defined to be
ρ(f) = lim
r→∞
log logT(r,f)
logr .
The hyper-exponent of convergence of zeros and distinct zeros off(z) are, respectively, defined to be (see [])
λ(f) = lim
r→∞
log logT(r,f)
logr , λ(f) =rlim→∞
log logT(r,f)
logr .
In addition, we useMto denote a positive constant, not necessarily the same at each oc-currence. We denote the linear measure of a setE⊂(, +∞) bymE=Edtand the loga-rithmic measure ofEbymlE=
Edt/t, respectively. The upper and the lower logarithmic
density ofEare defined by
log densE= lim
r→∞
ml(E∩[,r])
logr , log densE=rlim→∞
ml(E∩[,r])
logr .
For the second order linear differential equation
f+A(z)f+B(z)f = , (.)
whereA(z) andB(z)≡ are entire functions of finite order, it is well known that every solutionf ≡ of (.) is of infinite order ifρ(A) <ρ(B) orρ(B) <ρ(A)≤/ (see [–]). For the case of ρ(A) > / andρ(B) <ρ(A), the possibility of solutions of infinite order of (.) remains open, many authors have studied the problem (e.g.see [–]). In , Laine and Wu proved the following.
Theorem A(see []) Suppose thatρ(B) <ρ(A) <∞and that T(r,A)∼logM(r,A)as r→ ∞outside a set of finite logarithmic measure.Then every nonconstant solution f of
(.)is of infinite order.
For the higher order linear differential equation
f(k)+Ak–(z)f(k–)+· · ·+A(z)f =F(z), (.)
there are similar results as follows.
Theorem B (see []) Let Aj(z) (j= , . . . ,k– ), F(z)≡be entire functions. Suppose that there exists some d∈ {, . . . ,k– }such thatmax{ρ(F),ρ(Aj) :j=d}=ρ<ρ(Ad) <∞ and T(r,Ad)∼logM(r,Ad)as r→ ∞outside a set of upper logarithmic density less than
(ρ(Ad) –ρ)/ρ(Ad).Then every transcendental solution f(z)of(.)satisfiesλ(f) =λ(f) =
ρ(f) =∞.
Theorem C(see []) Let Aj(z) (j= , . . . ,k– ),F(z)≡be entire functions.Suppose that there exists some d∈ {, . . . ,k– }such thatmax{ρ(Aj) :j= , d}<ρ(A)≤and that Ad(z) has a finite deficient value.Then every solution f(z)≡of(.)satisfiesρ(A)≤ρ(f)≤ ρ(Ad).
Then a natural question is: Can we estimate the hyper-order of the solutions of (.) and (.) under the same condition in Theorems A and B? And: Can we estimate the hyper-order of the solutions of (.) in Theorem C ifρ(A) >? Theorems . and . below give answers to the above questions.
At the same time, many authors have investigated the growth of solutions of (.) and its non-homogeneous linear differential equation
f+A(z)f+B(z)f=F(z), (.)
whenρ(A) =ρ(B) and obtained the following results.
Theorem D(see []) Let P(z)and Q(z)be nonconstant polynomials such that P(z) =anzn+ an–zn–+· · ·+az+a,Q(z) =bnzn+bn–zn–+· · ·+bz+bfor some complex numbers
ai,bi(i= , . . . ,n)with anbn= and let A(z)and A(z)≡be entire functions satisfying
ρ(A) <n andρ(A) <n.Then the following statements hold:
(i) If eitherargan=argbnoran=cbnwith <c< ,then every nonconstant solutionf
of
f+A(z)eP(z)f+A(z)eQ(z)f = (.)
(ii) Letan=bnanddeg(P–Q) =m≥,and let the orders ofA(z)andA(z)be less thanm.Then every nonconstant solutionf of(.)has infinite order withρ(f)≥m. (iii) Letan=cbnwithc> anddeg(P–cQ) =m≥.Supposeρ(A) <mand
<ρ(A) <.Then every nonconstant solution of(.)has infinite order with ρ(f)≥ρ(A).
(iv) Letan=cbnwithc> and letP–cQbe a constant.Suppose thatρ(A) <ρ(A) <.
Then every nonconstant solution of(.)has infinite order withρ(f)≥ρ(A).
Theorem E(see []) Let a,b be nonzero complex numbers and a=b,Q(z)be a noncon-stant polynomial or Q(z) =h(z)ebz, where h(z)≡is a polynomial.Then every solution f ≡of the equation
f+eazf+Q(z)f= (.)
has infinite order andρ(f) = .
Theorem F(see []) Suppose that A≡,A≡,F are entire functions of order less
than one,and the complex constants a,b satisfy ab= and b=a.Then every nontrivial solution f of
f+A(z)eazf+A(z)ebzf=F(z) (.)
is of infinite order.
Theorem G(see []) Let P(z) =anzn+· · ·+a,Q(z) =bnzn+· · ·+bbe polynomials of
degree n≥where ai,bi(i= , , . . . ,n)are complex numbers,and let A(z)≡,A(z)≡,
F(z)be entire functions with order less than n.If an=bn,then every solution f ≡of
f+A(z)eP(z)f+A(z)eQ(z)f =F(z) (.)
is of infinite order.Furthermore,if F(z)≡,then every solution f of(.)satisfiesλ(f) = λ(f) =ρ(f) =∞.
Theorem D left us a question: Can we haveρ(f) =n(nis a positive integer) for every nontrivial solution of (.) ifan=bn? Theorem E tells us that the question holds ifn= . Many authors investigated the above question but none of them solve the question com-pletely, and Theorem . completely solves this question. In the following, we give our results.
Theorem . Let Aj (j= , . . . ,k– ),F(z)be entire functions.Suppose that there exists some d∈ {, . . . ,k– }such thatmax{ρ(Aj),ρ(F) :j= d} ≤ρ(Ad) <∞,max{τ(Aj) :ρ(Aj) = ρ(Ad),τ(F)}<τ(Ad)and that T(r,Ad)∼logM(r,Ad)as r→ ∞outside a set of r of finite logarithmic measure.Then we have:
(i) Every transcendental solutionf of(.)satisfiesρ(f) =ρ(Ad),and(.)may have
polynomial solutionsf of degree<d.
(iii) Ifd= ,then every nonconstant solutionf of(.)satisfiesρ(f) =ρ(A).
Furthermore,ifF(z)≡,then every nonconstant solutionf of(.)satisfies λ(f) =λ(f) =ρ(f) =ρ(A).
Theorem . Let Aj(j= , . . . ,k– ),F(z)≡be entire functions satisfyingmax{ρ(Aj) :j=
,d}<ρ(A) <ρ(Ad) <∞.Suppose that T(r,A)∼logM(r,A)as r→ ∞outside a set of
r of finite logarithmic measure and that Adhas a finite deficient value.Then every solution f ≡of(.)satisfiesρ(A)≤ρ(f)≤ρ(Ad).
Theorem . Let P(z),Q(z),A(z),A(z),F(z)satisfy the hypotheses of TheoremG.Then we have:
() Ifan=bn,F(z)≡,then every solutionf ≡of(.)satisfiesρ(f) =n. () Ifan=cbn(c< ),F(z)≡,then every solutionf of(.)satisfies
λ(f) =λ(f) =ρ(f) =n.
Remark . Theorems . and . are improvements of Theorems A-C. Theorem . is an improvement of Theorems D, E and a supplement to Theorems F, G.
2 Lemmas
Lemma .(see [], p.) Let Aj(z) (j= , . . . ,k– ),F(z)be entire functions satisfying
max{ρ(Aj),ρ(F) :j= , . . . ,k– } ≤ρ<∞.Then every solution f of(.)satisfiesρ(f)≤ρ.
Lemma .(see []) Let f(z)be a transcendental meromorphic function,and letα> be a given constant.Then for any given constant and for any givenε> :
(i) There exist a constantB> and a setE⊂(,∞)having finite logarithmic measure such that,for allzsatisfying|z|=r∈/E,we have
ff((ji))((zz))
≤B
T(αr,f)
r (logr)
αlogT(αr,f)
j–i
(≤i<j). (.)
(ii) There exist a setH⊂[, π)that has linear measure zero and a constantB> that depends only onα,for anyθ∈[, π)\H,there exists a constantR=R(θ) > such that,for allzsatisfyingargz=θ and|z|=r>R,we have
ff((ji))((zz))
≤BT(αr,f)logT(αr,f)j–i (≤i<j). (.)
Remark . We useE⊂(,∞) to denote a set ofrof finite logarithmic measure through-out this paper, not necessarily the same at each occurrence.
Lemma .(see []) Let f(z)be a transcendental entire function,and let zr=reiθr be a point satisfying|f(zr)|=M(r,f),then there exists a constantδr> (depending on r)such that,for all z satisfying|z|=r∈/Eandargz=θ∈[θr–δr,θr+δr],we have
ff(j()z(z))
≤rj (j∈N). (.)
measure such that,for all r∈E,we have
logM(r,f) >βrρ. (.)
Lemma . Let f(z)be a transcendental entire function satisfying <ρ(f) =ρ<∞,τ(f) = τ > ,and T(r,f)∼logM(r,f)as r→ ∞outside a set of r of finite logarithmic measure.
Then for anyβ<τ,there exists a set E⊂(,∞)having infinite logarithmic measure and
a set H⊂[, π)with linear measure zero such that,for all z satisfying|z|=r∈Eand
argz=θ∈[, π)\H,we have
freiθ >expβrρ. (.)
Proof Sincem(r,f)∼logM(r,f) asr→ ∞(r∈/ E), by the definition ofm(r,f), we see that there exists a setH⊂[, π) with linear measure zero such that for allzsatisfying
argz=θ∈[, π)\Hand for anyε> , we have
freiθ >M(r,f)–ε (r∈/E). (.)
Otherwise, we find that there exists a set H⊂[, π) with positive linear measure,i.e.,
mH> such that, for allzsatisfyingargz=θ∈Hand for anyε> , one has freiθ ≤M(r,f)–ε (r∈/E
).
Then, for allr∈/E, we have
m(r,f) = π
π
log+freiθ dθ
= π
H
log+freiθ dθ+ π
[,π)\H
log+freiθ dθ
≤ ( –ε)mH
π logM(r,f) +
π–mH
π logM(r,f) = π–ε·mH
π logM(r,f). (.)
Sinceε> ,mH> , (.) is a contradiction withm(r,f)∼logM(r,f).
For anyβ<τ, we chooseβsatisfyingβ<β<τ, by Lemma ., there exists a setE⊂ (,∞) having infinite logarithmic measure such that, for all|z|=r∈E, we have
M(r,f) >expβrρ
. (.)
By (.) and (.), for any given <ε< –ββ
and, for allzsatisfying|z|=r∈E\Eand
argz=θ∈[, π)\H, we have
freiθ >M(r,f)–ε>exp( –ε)β
rρ
>expβrρ.
Remark . The following lemma is a special case of Lemma in [].
Lemma .(see []) Let f(z)be a transcendental entire function satisfying <ρ(f) =ρ< ∞and T(r,f)∼logM(r,f)as r→ ∞outside a set of r of finite logarithmic measure.Then for any givenε> ,there exists a set E⊂(,∞)with positive upper logarithmic density
and a set H⊂[, π)with linear measure zero such that,for all z satisfying r∈Eand
|z|=r∈/Eandargz=θ∈[θr–δr,θr+δr], we have
ff(d()z()z)
≤rd. (.)
By Lemma ., there exist a setH⊂[, π) having linear measure zero and a constant
B> such that, for allzsatisfyingargz=θ∈[θr–δr,θr+δr]\Hand for sufficiently large
r, we have ff((ji))((zz))
≤BT(r,f)k (≤i<j≤k). (.)
We chooseα,αsatisfyingmax{τ(Aj) :ρ(Aj) =ρ(Ad),τ(F)}<α<α<τ(Ad), since|f(z) –
f(zr)|<εand|f(zr)| → ∞asr→ ∞, for all sufficiently large|z|=r∈/ Eandargz=θ ∈ [θr–δr,θr+δr], we have
Aj(z)≤expαrρ(Ad)
(j=d), F(z)
f(z)
≤F(z)≤expαrρ(Ad)
. (.)
SinceT(r,Ad)∼logM(r,Ad) asr→ ∞(r∈/ E), by Lemma ., for anyα<τ(Ad), there
exist a setE⊂(,∞) having infinite logarithmic measure and a setH⊂[, π) with linear measure zero such that for allzsatisfying|z|=r∈Eandargz=θ ∈[θr–δr,θr+
δr]\H, we have
Ad(z)>expαrρ(Ad). (.)
Substituting (.)-(.) into (.), for allzsatisfying|z|=r∈E\Eandargz=θ∈[θr–
δr,θr+δr]\(H∪H), we have
expαrρ(Ad)
≤(k+ )BT(r,f)k·rd·expαrρ(Ad)
. (.)
From (.), we haveρ(f)≥ρ(Ad). Therefore every transcendental solutionf(z) of (.) satisfiesρ(f) =ρ(Ad). Iff(z) is a polynomial solution of (.) withdegf ≥d, then by a
simple estimation on both sides of (.), we haveρ(f(k)+A
k–f(k–)+· · ·+Af) =ρ(F) < ρ(Ad), this is a contradiction, therefore each polynomial solutionf of (.) must satisfy
degf <d.
(ii) IfF≡, by Lemma ., we have that every transcendental solutionf(z) of (.) sat-isfiesλ(f) =λ(f) =ρ(f) =ρ(Ad).
(iii) Ifd= , it is easy to see that (.) cannot have polynomial solutions, and by (i) and (ii), we see that every nonconstant solutionf(z) of (.) satisfiesρ(f) =ρ(A) andλ(f) =
λ(f) =ρ(f) =ρ(A) ifF≡.
Proof of Theorem. Suppose thatAd(z) hasa∈Cas a finite deficient value and satis-fyingδ(a,Ad) = β> . Then by the definition of deficiency, for sufficiently larger, we havem(r,
Ad–a) >βT(r,Ad). Hence, for sufficiently larger, there exists a pointzr=re iθr
satisfying|zr|=rand
Substituting (.), (.)-(.) into (.), for all|z|=r∈(Eξ ∩E)\E andargz=θ ∈ satisfiesρ(f)≤nand that (.) has no polynomial solutions by the assumption. In the following, we need to show that every transcendental solutionf of (.) satisfiesρ(f)≥n. We divide the proof into two parts: (i)argan=argbnoran=cbn( <c< ), (ii)an=cbn
that, for allzsatisfying|z|=r∈/Eandargz=θ∈[θr–δ,θr+δ]\H, we have
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1College of Information System and Management, National University of Defense Technology, Changsha, Hunan 410073,
China.2College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China.
Acknowledgements
This project is supported by the National Natural Science Foundation of China (11301232, 11301233, 11261024), the Natural Science Foundation of Jiangxi Province in China (20132BAB211002, 20151BAB201004) and the Foundation of Education Bureau of Jiangxi Province in China (GJJ14271, GJJ14272).
Received: 28 February 2015 Accepted: 10 July 2015
References
1. Hayman, WK: Meromorphic Functions. Clarendon, Oxford (1964)
2. Laine, I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993)
3. Yi, HX, Yang, CC: The Uniqueness Theory of Meromorphic Functions. Science Press, Beijing (1995) (in Chinese) 4. Chen, ZX, Yang, CC: Some further results on the zeros and growth of entire solutions of second order differential
equations. Kodai Math. J.22, 273-285 (1999)
5. Gundersen, G: Finite order solutions of second order linear differential equations. Trans. Am. Math. Soc.305, 415-429 (1988)
6. Hellerstein, S, Miles, J, Rossi, J: On the growth of solutions off+gf+hf= 0. Trans. Am. Math. Soc.324, 693-706 (1991)
8. Kwon, KH: Nonexistence of finite order solutions of certain second order linear differential equations. Kodai Math. J.
19, 378-387 (1996)
9. Kwon, KH, Kim, JH: Maximum modulus, characteristic, deficiency and growth of solution of second order linear differential equations. Kodai Math. J.24, 344-351 (2001)
10. Laine, I, Wu, PC: Growth of solutions of second order linear differential equations. Proc. Am. Math. Soc.128, 2693-2703 (2000)
11. Wu, PC, Zhu, J: On the growth of solutions to the complex differential equationf+A(z)f+B(z)f= 0. Sci. China Ser. A
54(5), 939-947 (2011)
12. Tu, J, Deng, GT: Growth of solutions of a class of higher order linear differential equation. Complex Var. Elliptic Equ.
53(7), 623-631 (2008)
13. Chen, ZX: The growth of solutions off+e–zf+Q(z)f= 0 where the order (Q) = 1. Sci. China Ser. A45(3), 290-300
(2002)
14. Wang, J, Laine, I: Growth of solutions of second order linear differential equations. J. Math. Anal. Appl.342, 39-51 (2008)
15. Belaïdi, B: Growth and oscillation related to a second order linear differential equation. Math. Commun.18(1), 171-184 (2013)
16. Kinnunen, L: Linear differential equations with solutions of finite iterated order. Southeast Asian Bull. Math.22(4), 385-405 (1998)
17. Gundersen, G: Estimates for the logarithmic derivate of a meromorphic function, plus similar estimates. J. Lond. Math. Soc.37, 88-104 (1988)
18. Tu, J, Xu, HY, Liu, HM, Liu, Y: Complex oscillation of higher order linear differential equations with coefficients being lacunary series of finite iterated order. Abstr. Appl. Anal.2013, Article ID 634739 (2013)
19. Tu, J, Yi, CF: On the growth of solutions of a class of linear differential equations with coefficients having the same order. J. Math. Anal. Appl.340(1), 487-497 (2008)
20. Fuchs, W: Proof of a conjecture of G. Pólya concerning gap series. Ill. J. Math.7, 661-667 (1963)
21. Cao, TB, Chen, ZX, Zheng, XM, Tu, J: On the iterated order of meromorphic solutions of higher order linear differential equations. Ann. Differ. Equ.21(2), 111-122 (2005)
