This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomials L(f)=d2f″+d1f′+d0f generated by solutions of the above equation, where d0(z),d1(z), and d2(z) are entire functions that are not all equal to zero.

1. Introduction and Main Results

A function f(z) is called meromorphic if it is nonconstant and analytic in the complex plane ℂ except at possible isolated poles. If no poles occur, then f(z) reduces to an entire function. Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna’s value distribution theory of meromorphic functions, for reference see [1]. In addition, we use notations σ(f) and λ(f) to denote the order and the exponent of convergence of zero sequence and λ¯(f) to denote the sequence of distinct zeros of f(z), respectively. A meromorphic function ψ(z) is called a small function with respect to f(z) if T(r,ψ)=o(T(r,f)) as r→∞, possibly outside of a set of r with finite measure, where T(r,f) is the Nevanlinna characteristic function of f(z).

For the second-order linear differential equation
f′′+e-zf′+B(z)f=0,
where B(z) is an entire function of finite order, it is well known that each solution f of (1.1) is an entire function, and that if f1 and f2 are any two linearly independent solutions of (1.1), then at least one of f1,f2 must have infinite order, see [2, pages 167-168].

Thus, a natural question is the following: what condition on B(z) will guarantee that every solution f≢0 of (1.1) has infinite order? Many researchers have studied the question, for the details see [3, page 291]. For the case that B(z) is a transcendental entire function, Gundersen [4] proved that if σ(B)≠1, then every solution f≢0 of (1.1) has infinite order. In 2002, Chen considered the problem and proved the following result which is an improvement of Gundersen’s result.

Theorem A (see [<xref ref-type="bibr" rid="B2">3</xref>]).

Let a, b be nonzero complex numbers satisfying ab≠0 and arga≠argb or a=cb(0<c<1), and let Aj(z)≢0(j=1,2) be entire functions with σ(Aj)<1, then every solution f(≢0) of the equation
f′′+A1(z)eazf′+A0(z)ebzf=0
has infinite order.

Some further results on (1.2) were obtained for several cases. Chen [3] got the same conclusion when a=cb(c>1), and Chen and Shon [5] investigated the more general equations with meromorphic coefficients. Under the same assumption of Theorem A, if A1(z) and A0(z) are meromorphic functions with σ(Aj)<1(j=0,1), then there is the same conclusion with Theorem A. In 2008, Wang and Laine [6] extended Theorem A to nonhomogeneous second-order linear differential equations.

Theorem B (see [<xref ref-type="bibr" rid="B14">6</xref>]).

Let Aj(z)≢0(j=0,1) and F≢0 be entire functions with max{σ(Aj),(j=0,1),σ(F)}<1, and let a,b be complex constants that satisfy ab≠0 and a≠b, then every solution f of differential equation
f′′+A1(z)eazf′+A0(z)ebzf=F
is of infinite order.

Remark 1.1.

Belaïdi and El Farissi [7] also proved Theorem B and got λ¯(f)=λ(f)=σ(f)=∞. We note that (2.21) in [7] cannot be deduced by following their proof. Indeed, as r→∞, |f(z)|>1 holds just for the points z satisfying |f(z)|=M(r,f), not for all z. However, the difficulty can be got over by using Lemmas 2.5 and 2.6 in [8], and the method can be used in our proof of the following Theorem 1.2.

Since the beginning of the last four decades, a substantial number of research papers have been written to describe the fixed points of general transcendental functions. However, there are few studies on the fixed points of solutions of the general differential equations. In 2000, Chen [9] first studied the problems on the fixed points of solutions of second-order linear differential equations with entire coefficients. Since then, many results on fixed points of solutions of differential equations with entire coefficients were obtained, see [10–12]. In 2006, Chen and Shon [13] further studied the relation between small functions and solutions or differential polynomials of solutions of differential equations and obtained the following.

Theorem C.

Let Aj(z)≢0(j=0,1) be entire functions with σ(Aj)<1, and let a,b be complex constants such that ab≠0 and arga≠argb or a=cb(0<c<1). If ψ(z)≢0 is an entire function with finite order, then every solution f≢0 of (1.2) satisfies λ¯(f-ψ)=λ¯(f′-ψ)=λ¯(f′′-ψ)=∞. Furthermore, let d0(z),d1(z),and d2(z) be polynomials that are not all equal to zero, and let L(f)=d2f′′+d1f′+d0f. If the order of ψ is less than 1, then λ¯(L(f)-ψ)=∞.

Belaïdi and El Farissi [7] also studied the relation between small functions and some differential polynomials generated by solutions of the second-order nonhomogeneous linear differential equation (1.3). They obtained the following.

Theorem D.

Let Aj(z)≢0(j=0,1) and F≢0 be entire functions with max{σ(Aj)(j=0,1),σ(F)}<1, and let a,b be complex constants that satisfy ab≠0 and arga≠argb or a=cb(0<c<1). Let d0(z),d1(z),d2(z) be entire functions that are not all equal to zero with σ(dj)<1(j=0,1,2), and let ψ(z) be an entire function with finite order. If f is a solution of (1.3), then the differential polynomial L(f)=d2f′′+d1f′+d0f satisfies λ¯(L(f)-ψ)=∞.

The main purpose of this paper is to study the growth and the oscillation of solutions of second-order linear differential equation with meromorphic coefficients. Also, we will investigate the relation between small functions and differential polynomials generated by solutions of the above equation. Our results can be stated as follows.

Theorem 1.2.

Let Aj(z)≢0 (j=0,1) and F(z) be meromorphic functions with max{σ(F),σ(Aj)}<n, and let P(z)=anzn+⋯+a0, Q(z)=bnzn+⋯+b0 be polynomials with degree n (n≥1), where ai, bi (i=0,1,…,n), anbn≠0 are complex constants such that argan≠argbn or an=cbn(0<c<1), then every meromorphic solution f≢0 of the equation
f′′+A1eP(z)f′+A0eQ(z)f=F
has infinite order and satisfies
λ¯(f)=λ(f)=σ(f)=∞.

Theorem 1.3.

Under the assumption of Theorem 1.2, and let d0(z),d1(z),d2(z) be meromorphic functions that are not all equal to zero with σ(dj)<1(j=0,1,2), and let ψ(z) be a meromorphic function with finite order, if f≢0 is a meromorphic solution of (1.4), then the differential polynomial L(f)=d2f′′+d1f′+d0f satisfies λ¯(L(f)-ψ)=∞.

Remark 1.4.

Clearly, the method used in linear differential equations with entire coefficients cannot deal with the case of meromorphic coefficients. In addition, the proof of the results in [7, 13] relies heavily on the idea of Lemma 5 in [13] or Lemma 2.5 in [7]. However, it seems too complicated to deal with our cases. We will use an important result in uniqueness theory of meromorphic functions, that is Lemma 2.5, to prove our theorems.

2. Preliminary Lemmas

In order to prove our theorems, we need the following lemmas.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B8">14</xref>]).

Let w(z) be a transcendental meromorphic function with σ(f)=σ<∞. Let Γ={(k1,j1),…,(km,jm)} be a finite set of distinct pairs of integers satisfying ki>ji≥0 for i=1,2,…,m. Also let ϵ>0 be a given constant, then there exists a set E1⊂(1,∞) that has finite logarithmic measure, such that for all z satisfying z∉E∪[0,1] and for all (k,j)∈Γ, one has
|w(k)(z)||w(j)(z)|≤|z|(k-j)(σ-1+ɛ).

Now we introduced a notation, see [15] and [8, Lemma 2.3]. Let P(z)=(α+βi)zn+⋯ is a nonconstant polynomial, and α,β is real constants. For θ∈[0,2π), set δ(P(z),θ)=αcosnθ-βsinnθ.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B13">15</xref>]).

Let P(z) be a nonconstant polynomial of degree n. Let w(z) be a meromorphic function, not identically zero, of order less than n, and set g(z)=w(z)eP(z). Then for any given ɛ>0 there exists a zero measure set H1⊂[0,2π) such that if θ∈θ∈[0,2π)∖(H1∪H2), then for |z|>r(θ),

if δ(P,θ)<0, then exp((1+ϵ)δ(P,θ)rn)≤|g(reiθ)|≤exp((1-ϵ)δ(P,θ)rn),

if δ(P,θ)>0, then exp((1-ϵ)δ(P,θ)rn)≤|g(reiθ)|≤exp((1+ϵ)δ(P,θ)rn), where H2={θ:δ(P,θ)=0,0≤θ<2π} is a finite set.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B16">8</xref>, Lemma 2.5]).

Let f(z) be an entire function, and suppose that
G(z):=log+|f(k)||z|ρ
is unbounded on some ray argz=θ with constant ρ>0, then there exists an infinite sequence of points zn=rneiθ(n=1,2,…), where rn→∞, such that G(zn)→∞ and
|f(j)(zn)||f(k)(zn)|≤1(k-j)!(1+o(1))rnk-j,j=0,…,k-1,
as n→∞.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B5">16</xref>]).

Let A0,…,Ak-1,F≢0 be finite-order meromorphic functions. If f is an infinite-order meromorphic solution of the equation
f(k)+Ak-1f(k-1)+⋯+A0f=F,
then f satisfies λ(f)=λ¯(f)=σ(f)=∞.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B17">17</xref>, page 79]).

Suppose that f1(z),f2(z),…,fn(z)(n≥2) are meromorphic functions and g1(z),g2(z),…,gn(z) are entire functions satisfying the following conditions:

Σj=1nfj(z)egj(z)≡0,

gj(z)-gk(z) are not constants for 1≤j<k≤n,

for 1≤j≤n,1≤h<k≤n, T(r,fj)=o{T(r,egh-gk)}(r→∞,r∉E), where E has a finite measure.

then fj(z)≡0(j=1,2,…,n).Lemma 2.6 (see [<xref ref-type="bibr" rid="B16">8</xref>, Lemma 2.6]).

Let f(z) be a an entire function of order σ(f)=σ<∞. Suppose that there exists a set E⊂[0,2π) which has linear measure zero, such that log+|f(reiθ)|≤Mrρ for any ray argz=θ∈[0,2π)∖E, where M is a positive constant depending on θ, while ρ is a positive constant independent of θ. Then σ(f)≤ρ.

Lemma 2.7.

Under the assumption of Theorem 1.2, and let f be a meromorphic solution of (1.4). Set w=f′′+A1eP(z)f′+A0eQ(z)f. If f≢0 is of finite order, then σ(w)≥n.

Proof.

Suppose the contrary that σ(w)<n, we will deduce a contradiction.

First, if f(z)≡C≠0, then w=CA0eQ(z). Clearly, σ(w)=n, this is a contradiction.

Now suppose that f≢C. If σ(f)<n, then
f′′+A1eP(z)f′+A0eQ(z)f-w=0.
By Lemma 2.5, we have A0≡0, and this is a contradiction. Hence, σ(f)≥n.

Since f is a meromorphic solution of (1.4), we know that the poles of f can occur only at the poles of Aj(j=0,1) and F. Let f=g(z)/d(z), where d(z) is the canonical product formed with the nonzero poles of f(z), with σ(d)≤max{σ(F),σ(Aj),j=0,1}<n, and g is an entire function with n≤σ(g)=σ(f)=σ≤∞. Substituting f=g/d into (2.5), by some calculation we can get
dw=g′′+g′[A1eP(z)-2(d′d)]+g[A0eQ(z)-A1eP(z)d′d+2(d′d)2-d′′d].
Now, we rewrite (2.6) into
dwg-g′′g-[A1eP(z)-2(d′d)]g′g=[A0eQ(z)-A1eP(z)d′d+2(d′d)2-d′′d].

Set max{σ(w),σ(Aj),j=0,1}=β<n. By Lemma 2.1, for any given ɛ(0<ɛ<1-β), there exists a set E2∈[0,2π) which has linear measure zero, such that if θ∈[0,2π)∖E2, then there is a constant R1=r1(θ)>1 such that for all z satisfying argz=θ and |z|≥R1, we have
|g(i)(z)||g(z)|≤|z|2(σ-1+ɛ),|d(i)(z)||d(z)|≤|z|2(β-1+ɛ),i=1,2.

Case 1.

Suppose that an=cbn(0<c<1), then by Lemmas 2.1 and 2.2, there exists a ray argz=θ∈[0,2π)∖E2∪H1∪H2, H1 and H2 being defined in Lemma 2.2, such that δ(P,θ)=cδ(Q,θ)>0, and for the above ɛ and sufficiently r,
|A0eQ(z)-A1eP(z)d′d+2(d′d)2-d′′d|≥|A0eQ(z)|-|A1eP(z)||d′d|-|2(d′d)2|-|d′′d|≥12exp{(1-ɛ)δ(Q,θ)rn}.
Also, by Lemmas 2.1 and 2.2, we have
|A1eP(z)-2(d′d)|≤|A1eP(z)|+|2d′d|≤Mexp{(1+ɛ)cδ(Q,θ)rn},
where M is a constant.

Now we claim that
log+|g(z)||z|β+ɛ
is bounded on the ray argz=θ. Otherwise, by Lemma 2.3, there exists a sequence of points zm=rmeiθ, such that rm→∞log+|g(zm)|rmβ+ɛ⟶∞.
From (2.12) and the definition of order, we see that
|d(zm)w(zm)g(zm)|⟶0,
for m is large enough. By (2.7), (2.8), (2.9), (2.10), and (2.13), we get
12exp{(1-ɛ)δ(Q,θ)rmn}≤|A0eQ(z)-A1eP(z)d′d+2(d′d)2-d′′d|≤|dwg|+|(A1eP(z)-2(d′d))g′g|+|g′′g|≤M1exp{(1+ɛ)cδ(Q,θ)rmn},
where M1 is a constant. Clearly, we can choose ɛ such that 0<ɛ<(1-c)/(1+c). Then by (2.14), we can obtain a contradiction. Therefore,
log+|g(z)||z|β+ɛ
is bounded, and we have |g(z)|≤Mexp{rβ+ɛ} on the ray argz=θ.

Case 2.

Suppose that argan≠argbn. By Lemma 2.2, there exists a ray argz=θ∈[0,2π)∖E2∪H1∪H2, where E2, H1, and H2 are defined, respectively, as in Case 1, such that
δ(Q,θ)>0,δ(P,θ)<0.

Then, for any given ɛ(0<ɛ<n-β), by Lemma 2.2 and (2.7), we have, for sufficiently large |z|=r,
|A0eQ(z)-A1eP(z)d′d+2(d′d)2-d′′d|≥|A0eQ(z)|-|A1eP(z)||d′d|-|2(d′d)2|-|d′′d|≥12exp{(1-ɛ)δ(Q,θ)rn},|A1eP(z)-2(d′d)|≤|A1eP(z)|+2|(d′d)|≤exp{β+ɛ}exp{(1-ɛ)δ(P,θ)rn}+r2(β-1+ɛ).
As in Case 1, we prove that
log+|g(z)||z|β+ɛ
is bounded on the ray argz=θ. Otherwise, similarly as in Case 1, there exists a sequence of points zm=rmeiθ, such that rm→∞,
log+|g(zm)|rmβ+ɛ⟶∞.
Further, we have
|d(zm)w(zm)g(zm)|⟶0,
for m is large enough.

By (2.7), (2.8), (2.17), (2.18), and (2.21), we get
12exp{(1-ɛ)δ(Q,θ)rmn}≤|A0eQ(z)-A1eP(z)d′d+2(d′d)2-d′′d|≤|dwg|+|(A1eP(z)-2(d′d))g′g|+|g′′g|≤exp{(1-ɛ)δ(P,θ)rmn}+rm2(β-1+ɛ).
Since δ(Q,θ)>0 and δ(P,θ)<0, we obtain a contradiction. So
log+|g(z)||z|β+ɛ
is bounded, and we have
|g(z)|≤Mexp{rβ+ɛ}
on the ray argz=θ.

Combining Cases 1 and 2, for any given ray argz=θ∈[0,2π)∖E, E of linear measure zero, we have (2.24) on the ray argz=θ, provided that r is sufficiently large. Thus by Lemma 2.6, we get σ(g)≤β+ɛ<n, which is a contradiction. Then σ(w)≥n.

Lemma 2.8.

Under the assumption of Theorem 1.3, let f(z) be an infinite-order meromorphic solution of (1.4), then σ(L(f))=∞.

Proof.

Suppose that f(z) is a meromorphic solution of (1.4), then by Theorem 1.2, we have σ(f)=∞.

Now suppose that d2≢0. Substituting f′′=F-A1eP(z)f′-A0eQ(z) into L(f), we have
L(f)-d2F=(d1-d2A1eP(z))f′+(d0-d2A0eQ(z))f.

Differentiating both sides of (2.25), and replacing f′′ with f′′=F-A1ePf′-A0eQf, we obtain
L(f)′-(d2F)′-(d1-d2A1eP)F=[d2A12e2P-((d2A1)′+P′d2A1+d1A1)eP-d2A0eQ+d0+d1′]f′+[d2A0A1eP+Q-((d2A0)′+Q′d2A0+d1A0)eQ+d0′]f.
Set
α1=d1-d2A1eP,α0=d0-d2A0eQ,β1=d2A12e2P-((d2A1)′+P′d2A1+d1A1)eP-d2A0eQ+d0+d1′,β0=d2A0A1eP+Q-((d2A0)′+Q′d2A0+d1A0)eQ+d0′.
Then we rewrite (2.25) and (2.26) into
α1f′+α0f=L(f)-d2F,β1f′+β0f=L(f)′-(d2F)′-(d1-d2A1eP)F.
Set
h=α1β0-α0β1=[d1-d2A1eP][d2A0A1eP+Q-((d2A0)′+Q′d2A0+d1A0)eQ+d0′]-[d0-d2A0eQ][d2A12e2P-((d2A1)′+P′d2A1+d1A1)eP-d2A0eQ+d0+d1′]=h0+hP(z)eP+hP+QeP+Q+hQeQ+h2Pe2P-d22A02e2Q,
where hi(z)(i∈Λ={0,P,Q,P+Q,2P}) are meromorphic functions formed by A0,A1,d0, and d1 and their derivatives, with order less than n, and Λ is a index set. Since any one of {P,Q,P+Q,2P} is not equal to 2Q, then by Lemma 2.5, we have d22A02≡0. This is a contradiction. Thus, h≢0.

By (2.28), we get
f=1h(α1(L(f)′-(d2F)′-α1F)-β1(L(f)-d2F)).
If σ(L(f))<∞, then by (2.30) we have σ(f)<∞. Clearly, it is a contradiction. Hence, σ(L(f))=∞.

Suppose that d2≡0,d1≢0 or d2=d1≡0, and d0≢0, then by similar discussion as above, we can get the same conclusion.

3. Proof of Theorem <xref ref-type="statement" rid="thm1">1.2</xref>

Let f≢0 be a meromorphic solution of (1.4). Conversely, suppose that σ(f)<∞. By Lemma 2.7, we have n≤σ(w)=σ(F)<n. This is a contradiction. By Lemma 2.4, f satisfies λ¯(f)=λ(f)=σ(f)=∞.

4. Proof of Theorem <xref ref-type="statement" rid="thm2">1.3</xref>

Suppose that argan≠argbn or an=cbn(0<c<1) and that f is a meromorphic solution of (1.4). Set k(z)=L(f)-ψ. By σ(ψ)<∞ and Lemma 2.8, we have σ(k)=∞. Without loss of generality, we assume that d2≢0. Indeed, the remaining cases can be obtained by similar discussion. Substituting L(f)=k(z)+ψ into (2.30), we havef=1h(α1(k′+ψ′-(d2F)′-α1F)-β1(k+ψ-d2F))=1h(α1k′-β1k)+ϕ,
whereϕ(z)=α1h(ψ′-(d2F)′-α1F)-β1h(ψ-d2F)
is a meromorphic function of finite order. Then substituting (4.1) into (1.4), we haveα1hk′′′+ϕ2k′′+ϕ1k′+ϕ0k=F-(ψ′′+A1ePψ′+A0eQψ),
where ϕj(j=0,1,2) are meromorphic functions formed by α1/h,β/h,ϕ, and their derivatives. If F-(ψ′′+A1ePψ′+A0eQψ)≡0, then by Theorem 1.2, we have σ(ψ)=∞. This is impossible, and hence F-(ψ′′+A1ePψ′+A0eQψ)≢0. Thus, by h≢0, α1≢0, (4.3), and Lemma 2.4, we get λ(k)=λ¯(k)=σ(k)=∞.

Acknowledgments

The author would like to thank the referees for their valuable suggestions and comments. This research was supported by NNSF of China (no. 11001057), NSF of Jiangsu Province (BK2010234), Project of Qinglan of Jiangsu Province.

HaymanW. K.HilleE.ChenZ.The growth of solutions of f″+e−zf′+Q(z)f=0 where the order σ(Q)=1GundersenG. G.On the question of whether f″+e−zf′+B(z)f=0 can admit a solution f≢0 of finite orderChenZ. X.ShonK. H.On the growth and fixed points of solutions of second order differential equations with meromorphic coefficientsWangJ.LaineI.Growth of solutions of second order linear differential equationsBelaïdiB.El FarissiA.Relation between differential polynomials and small functionsWangJ.LaineI.Growth of solutions of nonhomogeneous linear differential equationsChenZ. X.The fixed points and hyper order of solutions of second order complex differential equationsWangJ.LüW. R.The fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficientsLaineI.RieppoJ.Differential polynomials generated by linear differential equationsLiuM.-S.ZhangX.-M.Fixed points of meromorphic solutions of higher order linear differential equationsChenZ. X.ShonK. H.The relation between solutions of a class of second-order differential equations with functions of small growthGundersenG. G.Estimates for the logarithmic derivative of a meromorphic function, plus similar estimatesMarkushevichA. I.ChenZ. X.Zeros of meromorphic solutions of higher order linear differential equationsYangC.-C.YiH.-X.