International Journal of Analysis and Applications
ISSN 2291-8639
Volume 6, Number 2 (2014), 178-194
http://www.etamaths.com

GROWTH AND COMPLEX OSCILLATION OF LINEAR

DIFFERENTIAL EQUATIONS WITH MEROMORPHIC

COEFFICIENTS OF [p,q] −ϕ ORDER

RABAB BOUABDELLI AND BENHARRAT BELAÏDI∗

Abstract. This paper is devoted to considering the growth of solutions of
complex higher order linear differential equations with meromorphic coeffi-

cients under some assumptions for [p, q] − ϕ order and we obtain some results
which improve and extend some previous results of H. Hu and X. M. Zheng;
X. Shen, J. Tu and H. Y. Xu and others.

1. Introduction and main results

Throughout this paper, a meromorphic function will means meromorphic in the
whole complex plane. In this paper, we assume that readers are familiar with the
fundamental results and standard notations of the Nevanlinna’s theory of mero-
morphic functions (see [9, 18]).

Consider for n ≥ 2 the linear differential equations
(1.1) f(n) + An−1f

(n−1) + · · · + A1f′ + A0f = 0,

(1.2) f(n) + An−1f
(n−1) + · · · + A1f′ + A0f = F,

where A0, · · · ,An−1,F are meromorphic functions. In [11, 12], Juneja, Kapoor and
Bajpai investigated some properties of entire functions of [p,q]-order and obtained
some results concerning their growth. In [16], in order to maintain accordance with
general definitions of the entire function f of iterated p−order [13, 14], Liu-Tu-
Shi gave a minor modification of the original definition of the [p,q]-order given in
[11, 12]. By this new concept of [p,q]-order, the [p,q]-order of solutions of complex
linear differential equations (1.1) and (1.2) was investigated in the unit disc and in
the complex plane (see e.g. [2, 3, 4, 15, 16]). In [6] , I. Chyzhykov, J. Heittokangas
and J. Rättyä introduced the definition of ϕ−order of a meromorphic function in
the unit disc as follows.

Definition 1.1 ([6]) Let ϕ : [0, 1) → (0, +∞) be a non-decreasing unbounded
function, the ϕ−order of f in the unit disc is defined by

σ (f,ϕ) = lim sup
r→+∞

log+ T (r,f)

log ϕ (r)
,

2010 Mathematics Subject Classification. 34M10, 30D35.
Key words and phrases. Meromorphic functions, [p, q] − ϕ order, [p, q] − ϕ type, [p, q] − ϕ

exponent of convergence, differential equation.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

178



GROWTH AND COMPLEX OSCILLATION OF LINEAR DIFFERENTIAL EQUATIONS 179

and where in the following, T (r,f) is the characteristic function of Nevanlinna.

On the basic of Definition 1.1, recently in [17] , X. Shen, J. Tu and H. Y. Xu
introduced the new concept of [p,q] − ϕ order of meromorphic functions in the
complex plane to study the growth and zeros of second order linear differential
equations.

For all r ∈ R, we define exp1 r := er and expp+1 r := exp
(
expp r

)
, p ∈ N. We

also define for all r sufficiently large log1 r := log r and logp+1 r := log
(
logp r

)
,

p ∈ N. Moreover, we denote by exp0 r := r, log0 r := r, log−1 r := exp1 r and
exp−1 r := log1 r.

Definition 1.2 [17] Let ϕ : [0, +∞) → (0, +∞) be a non-decreasing unbounded
function, and p,q be positive integers and satisfy p ≥ q ≥ 1. Then the [p,q] − ϕ
order and [p,q]−ϕ lower order of a meromorphic function f are respectively defined
by

σ[p,q] (f,ϕ) = lim sup
r→+∞

logp T (r,f)

logq ϕ (r)
,

µ[p,q] (f,ϕ) = lim inf
r→+∞

logp T (r,f)

logq ϕ (r)
.

Definition 1.3 Let f be a meromorphic function satisfying 0 < σ[p,q] (f,ϕ) = σ <
∞. Then the [p,q] −ϕ type of f (z) is defined by

τ[p,q] (f,ϕ) = lim sup
r→+∞

logp−1 T (r,f)[
logq−1 ϕ (r)

]σ .
Definition 1.4 Let p,q be integers such that p ≥ q ≥ 1. Let f be a meromorphic
function satisfying 0 < µ[p,q] (f,ϕ) = µ < ∞. Then the lower [p,q] −ϕ type of f is
defined by

τ[p,q] (f,ϕ) = lim inf
r→+∞

logp−1 T (r,f)[
logq−1 ϕ (r)

]µ .
Definition 1.5 ([17]) Let f be a meromorphic function. Then, the [p,q] − ϕ
exponent of convergence of zero-sequence (distinct zero-sequence) of f is defined by

λ[p,q] (f,ϕ) = lim sup
r→+∞

logp n
(
r, 1
f

)
logq ϕ (r)

,

λ̄[p,q] (f,ϕ) = lim sup
r→+∞

logp n̄
(
r, 1
f

)
logq ϕ (r)

.

And the lower exponent of distinct zero-sequence of f is defined by

λ̄[p,q] (f,ϕ) = lim inf
r→+∞

logp n̄
(
r, 1
f

)
logq ϕ (r)

.

Remark 1.1. If ϕ (r) = r in the Definitions 1.2-1.5, then we obtain the standard
definitions of the [p,q]−order, [p,q]−type and [p,q]−exponent of convergence.



180 BOUABDELLI AND BELAÏDI

Remark 1.2 [17] Throughout this paper, we assume that ϕ : [0, +∞) → (0, +∞)
is a non-decreasing unbounded function and always satisfies the following two con-
ditions :
(i) lim

r→+∞
logp+1 r

logq ϕ(r)
= 0.

(ii) lim
r→+∞

logq ϕ(αr)

logq ϕ(r)
= 1 for some α > 1.

From Remark 1.2, we can obtain the following proposition.

Proposition 1.1 Suppose that ϕ (r) satisfies the condition (i) − (ii) .
a) If f (z) is a meromorphic function, then

λ[p,q] (f,ϕ) = lim sup
r→+∞

logp n
(
r, 1
f

)
logq ϕ (r)

= lim sup
r→+∞

logp N
(
r, 1
f

)
logq ϕ (r)

,

λ̄[p,q] (f,ϕ) = lim sup
r→+∞

logp n̄
(
r, 1
f

)
logq ϕ (r)

= lim sup
r→+∞

logp N
(
r, 1
f

)
logq ϕ (r)

.

b) If f (z) is a meromorphic function, then

λ̄[p,q] (f,ϕ) = lim inf
r→+∞

logp n̄
(
r, 1
f

)
logq ϕ (r)

= lim inf
r→+∞

logp N
(
r, 1
f

)
logq ϕ (r)

.

Proof. We prove only b), for the proof of a) see [17] . We have

N

(
r,

1

f

)
=

r∫
0

n̄
(
t, 1
f

)
− n̄

(
0, 1

f

)
t

dt + n̄

(
0,

1

f

)
log r.

It follows that for r > r0 > 1

N

(
r,

1

f

)
−N

(
r0,

1

f

)
=

r∫
0

n̄
(
t, 1
f

)
− n̄

(
0, 1

f

)
t

dt + n̄

(
0,

1

f

)
log r

−


r0∫

0

n̄
(
t, 1
f

)
− n̄

(
0, 1

f

)
t

dt + n̄

(
0,

1

f

)
log r0




=

r∫
r0

n̄
(
t, 1
f

)
− n̄

(
0, 1

f

)
t

dt + n̄

(
0,

1

f

)
(log r − log r0)

(1.3) =

r∫
r0

n̄
(
t, 1
f

)
t

dt ≤ n̄
(
r,

1

f

)
log

r

r0
.

Then by (1.3) and lim
r→+∞

logp+1 r

logq ϕ(r)
= 0, we obtain

lim inf
r→+∞

logp N
(
r, 1
f

)
logq ϕ (r)



GROWTH AND COMPLEX OSCILLATION OF LINEAR DIFFERENTIAL EQUATIONS 181

(1.4) ≤ max


lim infr→+∞

logp n̄
(
r, 1
f

)
logq ϕ (r)

, lim sup
r→+∞

logp+1 r

logq ϕ (r)


 = lim infr→+∞

logp n̄
(
r, 1
f

)
logq ϕ (r)

.

On the other hand, since α > 1, we have for r > 1

N

(
αr,

1

f

)
=

αr∫
0

n̄
(
t, 1
f

)
− n̄

(
0, 1

f

)
t

dt + n̄

(
0,

1

f

)
log αr

≥
αr∫
r

n̄
(
t, 1
f

)
− n̄

(
0, 1

f

)
t

dt + n̄

(
0,

1

f

)
log αr

≥
(
n̄

(
r,

1

f

)
− n̄

(
0,

1

f

))
log α + n̄

(
0,

1

f

)
log αr

(1.5) = n̄

(
r,

1

f

)
log α + n̄

(
0,

1

f

)
log r ≥ n̄

(
r,

1

f

)
log α.

By (1.5) and lim
r→+∞

logq ϕ(αr)

logq ϕ(r)
= 1, we get

lim inf
r→+∞

logp N
(
αr, 1

f

)
logq ϕ (αr)

≥ lim inf
r→+∞


logp n̄

(
r, 1
f

)
logq ϕ (r)

.
logq ϕ (r)

logq ϕ (αr)




≥ lim inf
r→+∞

logp n̄
(
r, 1
f

)
logq ϕ (r)

.lim inf
r→+∞

logq ϕ (r)

logq ϕ (αr)

(1.6) = lim inf
r→+∞

logp n̄
(
r, 1
f

)
logq ϕ (r)

.

By (1.4) and (1.6), it is easy to see that conclusion of b) holds.

Many authors have investigated complex oscillation properties of (1.1) and ob-
tained many results when the coefficients in (1.1) are entire or meromorphic func-
tions under some assumptions of [p,q]−order. Recently, Hu and Zheng investigated
the growth of solutions of (1.1) and obtained the following results.

Theorem A ([10]) Let p,q be integers such that p ≥ q > 1 or p > q = 1, and let
A0, · · · ,An−1 be meromorphic functions. Assume that λ[p,q]

(
1
A0

)
< µ[p,q] (A0) <

∞, and that max
{
σ[p,q] (Aj) ,j = 1, · · · ,n− 1

}
≤ µ[p,q] (A0) and

max
{
τ[p,q] (Aj) : σ[p,q] (Aj) = µ[p,q] (A0) ,j 6= 0

}
< τ[p,q] (A0) = τ.

If f (6≡ 0) is a meromorphic solution of (1.1) satisfying
N (r,f)

N (r,f)
< expp+1

{
b logq r

} (
b ≤ µ[p,q] (A0)

)
,

then we have

λ̄[p+1,q] (f −ψ) = µ[p+1,q] (f) = µ[p,q] (A0)
≤ σ[p,q] (A0) = σ[p+1,q] (f) = λ̄[p+1,q] (f −ψ) ,



182 BOUABDELLI AND BELAÏDI

where ψ (z) (6≡ 0) is a meromorphic function with σ[p+1,q] (ψ) < µ[p,q] (A0) .

Theorem B ([10]) Let p,q be integers such that p ≥ q > 1 or p > q = 1, and let
A0, · · · ,An−1 be meromorphic functions. Assume that λ[p,q]

(
1
A0

)
< µ[p,q] (A0) <

∞, and that max
{
σ[p,q] (Aj) ,j = 1, · · · ,n− 1

}
≤ µ[p,q] (A0) and

lim sup
r→+∞

n−1∑
j=1

m(r,Aj)
m(r,A0)

< 1. If f (6≡ 0) is a meromorphic solution of (1.1) satisfying

N(r,f)

N(r,f)
< expp+1

{
b logq r

} (
b ≤ µ[p,q] (A0)

)
, then we have

λ̄[p+1,q] (f −ψ) = µ[p+1,q] (f) = µ[p,q] (A0)

≤ σ[p,q] (A0) = σ[p+1,q] (f) = λ̄[p+1,q] (f −ψ) ,
where ψ (z) (6≡ 0) is a meromorphic function with σ[p+1,q] (ψ) < µ[p,q] (A0) .

For the case that the dominant coefficient A0 is replaced by an arbitrary coefficient
As (s ∈{1, · · · ,n− 1}), they obtained the following.

Theorem C ([10]) Let p,q be integers such that p ≥ q ≥ 1, and let A0, · · · ,An−1
be meromorphic functions. Suppose that there exists one As (0 ≤ s ≤ n − 1)
with λ[p,q]

(
1
As

)
< µ[p,q] (As) < ∞ and that max{σ[p,q] (Aj) , j 6= s} ≤ µ[p,q] (As)

and max{τ[p,q] (Aj) : σ[p,q] (Aj) = µ[p,q] (As) , j 6= s} < τ[p,q] (As) = τ. Then
every transcendental meromorphic solution f ( 6≡ 0) of (1.1) satisfying N(r,f)

N(r,f)
<

expp+1
{
b logq r

}
(b ≤ µ[p,q] (As)) satisfies µ[p+1,q] (f) ≤ µ[p,q] (As) ≤ µ[p,q] (f) and

σ[p+1,q] (f) ≤ σ[p,q] (As) ≤ σ[p,q] (f) . Moreover, every non-transcendental meromor-
phic solution f of (1.1) is a polynomial with degree deg (f) ≤ s− 1.

The main purpose of this paper is to make use of the concept of meromorphic
functions of [p,q] −ϕ-order to improve the results above.

Theorem 1.1 Let p,q be integers such that p ≥ q > 1 or p > q = 1, and let
A0, · · · ,An−1 be meromorphic functions.
Assume that λ[p,q]

(
1
A0
,ϕ
)
< µ[p,q] (A0,ϕ) < ∞, and that max{σ[p,q] (Aj,ϕ) ,j =

1, · · · ,n− 1} ≤ µ[p,q] (A0,ϕ) and max{τ[p,q] (Aj,ϕ) : σ[p,q] (Aj,ϕ) = µ[p,q] (A0,ϕ) ,
j 6= 0} < τ[p,q] (A0,ϕ) = τ, and where ϕ satisfies the conditions lim

r→+∞
logp+1 r

logq ϕ(r)
= 0

and lim
r→+∞

logq−1 ϕ(αr)

logq−1 ϕ(r)
= 1 for some α > 1. If f ( 6≡ 0) is a meromorphic solution of

(1.1) satisfying
N(r,f)

N(r,f)
< expp+1

{
b logq ϕ (r)

} (
b ≤ µ[p,q] (A0,ϕ)

)
, then we have

λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (A0,ϕ)

≤ σ[p,q] (A0,ϕ) = σ[p+1,q] (f,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) ,
where ψ (z) (6≡ 0) is a meromorphic function with σ[p+1,q] (ψ,ϕ) < µ[p,q] (A0,ϕ) .

Theorem 1.2 Let p,q be integers such that p ≥ q > 1 or p > q = 1, and let
A0, · · · ,An−1 be meromorphic functions.
Assume that λ[p,q]

(
1
A0
,ϕ
)
< µ[p,q] (A0,ϕ) < ∞, and that max{σ[p,q] (Aj,ϕ) ,j =



GROWTH AND COMPLEX OSCILLATION OF LINEAR DIFFERENTIAL EQUATIONS 183

1, · · · ,n−1}≤ µ[p,q] (A0,ϕ) and lim sup
r→+∞

n−1∑
j=1

m(r,Aj)
m(r,A0)

< 1, and where ϕ satisfies the

conditions (i) − (ii) of the Remark 1.2. If f (6≡ 0) is a meromorphic solution of
(1.1) satisfying

N(r,f)

N(r,f)
< expp+1

{
b logq ϕ (r)

} (
b ≤ µ[p,q] (A0,ϕ)

)
, then we have

λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (A0,ϕ)

≤ σ[p,q] (A0,ϕ) = σ[p+1,q] (f,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) ,
where ψ (z) (6≡ 0) is a meromorphic function with σ[p+1,q] (ψ,ϕ) < µ[p,q] (A0,ϕ) .

Theorem 1.3 Let p,q be integers such that p ≥ q ≥ 1, and let A0, · · · ,An−1
be meromorphic functions. Suppose that there exists one As (0 ≤ s ≤ n− 1) with
λ[p,q]

(
1
As
,ϕ
)
< µ[p,q] (As,ϕ) < ∞ and that max

{
σ[p,q] (Aj,ϕ) ,j 6= s

}
≤ µ[p,q] (As,ϕ)

and max
{
τ[p,q] (Aj,ϕ) : σ[p,q] (Aj,ϕ) = µ[p,q] (As,ϕ) , j 6= s

}
< τ [p,q] (As,ϕ) = τ,

and where ϕ satisfies the conditions (i) − (ii) of the Remark 1.2. Then every tran-
scendental meromorphic solution f (6≡ 0) of (1.1) satisfying
N(r,f)

N(r,f)
< expp+1

{
b logq ϕ (r)

} (
b ≤ µ[p,q] (As,ϕ)

)
satisfies µ[p+1,q] (f,ϕ) ≤ µ[p,q] (As,ϕ) ≤ µ[p,q] (f,ϕ) and σ[p+1,q] (f,ϕ) ≤ σ[p,q] (As,ϕ)
≤ σ[p,q] (f,ϕ) . Moreover, every non-transcendental meromorphic solution f (z) of
(1.1) is a polynomial with degree deg (f) ≤ s− 1.

Remark 1.3. If we put ϕ (r) = r in the Theorems 1.1, 1.2, 1.3, then we obtain
Theorems A, B, C.

2. Auxiliary lemmas

We need the following lemmas to obtain our results.

Lemma 2.1 ([5]) Let f be a meromorphic solution of (1.1) assuming that not
all coefficients Aj (z) are constants. Given a real constant γ > 1, and denoting

T (r) =
n−1∑
j=0

T (r,Aj) , we have

log m (r,f) < T (r){(log r) log T (r)}γ , if p = 0

and

log m (r,f) < r2p+γ−1T (r){log T (r)}γ , if p > 0,
outside of an exceptional set Ep with

∫
Ep

tp−1dt < +∞.

Remark 2.1. Especially, if p = 0, then the exceptional set E0 has finite logarith-
mic measure

∫
E0

dt
t

= mlE0.

Lemma 2.2 ([1] , [8]) Let g : [0, +∞) → R, h : [0, +∞) → R be monotone increasing
functions. If (i) g (r) ≤ h (r) outside of an exceptional set of finite linear measure,
or (ii) g (r) ≤ h (r) , r /∈ E1 ∪(0, 1], where E1 ⊂ [1,∞) is a set of finite logarithmic
measure, then for any β > 1, there exists r0 = r0 (β) > 0 such that g (r) ≤ h (βr)
for all r > r0.



184 BOUABDELLI AND BELAÏDI

Lemma 2.3 ([9]) Let f be a transcendental meromorphic function and n ≥ 1 be
an integer. Then

m

(
r,
f(n)

f

)
= O (log (rT (r,f)))

outside of a possible exceptional set E2 of r of finite linear measure, and if f is of
finite order of growth, then

m

(
r,
f(n)

f

)
= O (log r) .

Lemma 2.4 Let p,q be integers such that p ≥ q ≥ 1, and let f be a meromorphic
function satisfying µ[p,q] (f,ϕ) = µ < ∞

(
σ[p,q] (f,ϕ) = σ < ∞

)
, where ϕ (r) only

satisfies lim
r→+∞

logq ϕ(αr)

logq ϕ(r)
= 1 for some α > 1. Then there exists a set E3 ⊂ (1,∞)

of infinite logarithmic measure such that for all r ∈ E3, we have

µ = lim
r→+∞
r∈E3

logp T (r,f)

logq ϕ (r)
,


σ = lim

r→+∞
r∈E3

logp T (r,f)

logq ϕ (r)




and for any given ε > 0 and sufficiently large r ∈ E3
T (r,f) < expp

{
(µ + ε) logq ϕ (r)

} (
T (r,f) > expp

{
(σ −ε) logq ϕ (r)

})
.

Proof. We prove only the first assumption, for the second we use the same proof.
By the Definition 1.2, there exists an increasing sequence {rn}

∞
n=1 tending to ∞

satisfying
(

1 + 1
n+1

)
rn < rn+1 and

µ = µ[p,q] (f,ϕ) = lim
rn→∞

logp T (rn,f)

logq ϕ (rn)
.

Then for any given ε > 0, there exists an n1 such that for n ≥ n1 and any r ∈[
rn,
(
1 + 1

n

)
rn
]
, we have

logp T (r,f)

logq ϕ (r)
≤

logp T
((

1 + 1
n

)
rn,f

)
logq ϕ

((
1 + 1

n

)
rn
) logq ϕ((1 + 1n)rn)

logq ϕ (rn)
.

When q ≥ 1, we have
logq ϕ((1+

1
n )rn)

logq ϕ(rn)
→ 1 (n → +∞). Let

E3 =

∞⋃
n=n1

[
rn, (1 +

1

n
)rn

]
,

for any given ε > 0 and all r ∈ E3, we have

lim
r→+∞
r∈E3

logp T (r,f)

logq ϕ (r)
≤ lim
rn→∞

logp T
((

1 + 1
n

)
rn,f

)
logq ϕ

((
1 + 1

n

)
rn
) = µ[p,q] (f,ϕ) ,

where mlE3 =
∞∑

n=n1

(1+ 1n )rn∫
rn

dt
t

=
∞∑

n=n1

log
(
1 + 1

n

)
= ∞. On the other hand, we

have

lim
r→+∞
r∈E3

logp T (r,f)

logq ϕ (r)
≥ lim inf

r→+∞

logp T (r,f)

logq ϕ (r)
= µ[p,q] (f,ϕ) .



GROWTH AND COMPLEX OSCILLATION OF LINEAR DIFFERENTIAL EQUATIONS 185

Therefore,

lim
r→+∞
r∈E3

logp T (r,f)

logq ϕ (r)
= µ[p,q] (f,ϕ)

and for any given ε > 0 and sufficiently large r ∈ E3

T (r,f) < expp
{

(µ + ε) logq ϕ (r)
}
.

Lemma 2.5 Let f1,f2 be meromorphic functions of [p,q] − ϕ order satisfying
σ[p,q] (f1,ϕ) > σ[p,q] (f2,ϕ) , where ϕ (r) only satisfies lim

r→+∞
logq ϕ(αr)

logq ϕ(r)
= 1 for some

α > 1. Then there exists a set E4 ⊂ (1, +∞) having infinite logarithmic measure
such that for all r ∈ E4, we have

lim
r→+∞

T (r,f2)

T (r,f1)
= 0.

Proof. Set σ1 = σ[p,q] (f1,ϕ) , σ2 = σ[p,q] (f2,ϕ) (σ1 > σ2) . By Lemma 2.4, there
exists a set E4 ⊂ (1, +∞) having infinite logarithmic measure such that for any
given 0 < ε < σ1−σ2

2
and all sufficiently large r ∈ E4

T (r,f1) > expp
{

(σ1 −ε) logq ϕ (r)
}

and for all sufficiently large r

T (r,f2) < expp
{

(σ2 + ε) logq ϕ (r)
}
.

From this we can get

T (r,f2)

T (r,f1)
<

expp
{

(σ2 + ε) logq ϕ (r)
}

expp
{

(σ1 −ε) logq ϕ (r)
}

=
1

exp
{

expp−1
{

(σ1 −ε) logq ϕ (r)
}
− expp−1

{
(σ2 + ε) logq ϕ (r)

}}, r ∈ E4.
Since 0 < ε < σ1−σ2

2
, then we have

lim
r→+∞

T (r,f2)

T (r,f1)
= 0, r ∈ E4.

Remark 2.2 If µ[p,q] (f1,ϕ) > µ[p,q] (f2,ϕ) , then we get the same result.

Lemma 2.6 Let p,q be integers such that p ≥ q ≥ 1, and let A0, · · · ,An−1, F ( 6≡ 0)
be meromorphic functions. If f is a meromorphic solution of (1.2) satisfying

max
{
σ[p,q] (F,ϕ) ,σ[p,q] (Aj,ϕ) ,j = 0, · · · ,n− 1

}
< µ[p,q] (f,ϕ) ,

then we have

λ̄[p,q] (f,ϕ) = λ[p,q] (f,ϕ) = µ[p,q] (f,ϕ) ,

where ϕ satisfies the conditions (i) − (ii) of Remark 1.2.

Proof. By (1.2) , we get

(2.1)
1

f
=

1

F

(
f(n)

f
+ An−1

f(n−1)

f
+ · · · + A1

f′

f
+ A0

)
.



186 BOUABDELLI AND BELAÏDI

It is easy to see that if f has a zero at z0 of order α (α > n) , and A0, · · · ,An−1 are
analytic at z0, then F must have a zero at z0 of order α−n. Hence

(2.2) N

(
r,

1

f

)
≤ nN

(
r,

1

f

)
+ N

(
r,

1

F

)
+

n−1∑
j=0

N (r,Aj) .

By the Lemma 2.3 and (2.1), we have

(2.3) m

(
r,

1

f

)
≤ m

(
r,

1

F

)
+

n−1∑
j=0

m (r,Aj) + O (log T (r,f) + log r) (r /∈ E2) ,

where E2 ⊂ (1, +∞) is a set of r of finite linear measure. By (2.2) and (2.3) , we
get

T (r,f) = T

(
r,

1

f

)
+ O (1) ≤ nN

(
r,

1

f

)
+ T (r,F)

(2.4) +

n−1∑
j=0

T (r,Aj) + O{log (rT (r,f))} (r /∈ E2) .

Since max{σ[p,q] (F,ϕ) , σ[p,q] (Aj,ϕ) , j = 0, · · · ,n− 1} < µ[p,q] (f,ϕ) , then

(2.5) max

{
T (r,F)

T (r,f)
,
T (r,Aj)

T (r,f)
( j = 0, · · · ,n− 1)

}
→ 0, r → +∞.

Also, for all sufficiently large r, we have

(2.6) log (T (r,f)) = o{T (r,f)} .

By (2.4) − (2.6) , for all |z| = r /∈ E2, we have

(2.7) (1 −o (1)) T (r,f) ≤ nN
(
r,

1

f

)
+ O (log r) .

By Definition 1.2, Proposition 1.1, Lemma 2.2 and (2.7) , we get

(2.8) µ[p,q] (f,ϕ) ≤ λ̄[p,q] (f,ϕ) .

Since µ[p,q] (f,ϕ) ≥ λ[p,q] (f,ϕ) ≥ λ̄[p,q] (f,ϕ) , then by (2.8) , we have

λ̄[p,q] (f,ϕ) = λ[p,q] (f,ϕ) = µ[p,q] (f,ϕ) .

Using the same method above, Lemma 2.5 and Lemma 2.2 we can prove the
following lemma.

Lemma 2.7 Let p,q be integers such that p ≥ q ≥ 1, and let A0, · · · ,An−1, F ( 6≡ 0)
be meromorphic functions. If f is a meromorphic solution of (1.2) satisfying

max
{
σ[p,q] (F,ϕ) ,σ[p,q] (Aj,ϕ) ,j = 0, · · · ,n− 1

}
< σ[p,q] (f,ϕ) < +∞,

then we have

λ[p,q] (f,ϕ) = λ[p,q] (f,ϕ) = σ[p,q] (f,ϕ) ,

where ϕ satisfies the conditions (i) − (ii) of Remark 1.2.

Lemma 2.8 Let p,q be integers such that p ≥ q ≥ 1 and let A0, · · · ,An−1 be
meromorphic functions such that max

{
σ[p,q] (Aj,ϕ) : j 6= s

}
≤ µ[p,q] (As,ϕ) < ∞,

where ϕ satisfies the conditions (i)−(ii) of Remark 1.2. If f ( 6≡ 0) is a meromorphic



GROWTH AND COMPLEX OSCILLATION OF LINEAR DIFFERENTIAL EQUATIONS 187

solution of (1.1) satisfying
N(r,f)

N(r,f)
< expp+1

{
b logq ϕ (r)

} (
b ≤ µ[p,q] (As,ϕ)

)
, then

we have

µ[p+1,q] (f,ϕ) ≤ µ[p,q] (As,ϕ) .
Proof. By (1.1), we know that the poles of f can only occur at the poles of

A0, · · · ,An−1. By
N(r,f)

N(r,f)
< expp+1

{
b logq ϕ (r)

} (
b ≤ µ[p,q] (As,ϕ)

)
, we have

N (r,f) < expp+1
{
b logq ϕ (r)

}
N (r,f) ≤ expp+1

{
b logq ϕ (r)

}n−1∑
j=0

N (r,Aj)

(2.9) ≤ expp+1
{
b logq ϕ (r)

}n−1∑
j=0

T (r,Aj) .

Then by (2.9) , we have

(2.10) T (r,f) ≤ m (r,f) + expp+1
{
b logq ϕ (r)

}n−1∑
j=0

T (r,Aj) .

By Lemma 2.4, there exists a set E3 of infinite logarithmic measure such that for
any given ε > 0 and sufficiently large r ∈ E3, we have

(2.11) T (r,As) ≤ expp
{(
µ[p,q] (As,ϕ) + ε

)
logq ϕ (r)

}
.

Since max
{
σ[p,q] (Aj,ϕ) : j 6= s

}
≤ µ[p,q] (As,ϕ) , for the above ε > 0 and suffi-

ciently large r, we have

(2.12) T (r,Aj) ≤ expp
{(
µ[p,q] (As,ϕ) + ε

)
logq ϕ (r)

}
, j 6= s.

By (2.11) , (2.12), Lemma 1.1 and Remark 1.2, there exists a set E0 of r of finite
logarithmic measure such that for sufficiently large r ∈ E3�E0

m (r,f) ≤ exp



n−1∑
j=0

T (r,Aj)


(log r) log


n−1∑
j=0

T (r,Aj)




γ



(2.13) ≤ expp+1
{(
µ[p,q] (As,ϕ) + 2ε

)
logq ϕ (r)

}
.

From (2.10) and (2.13) , we get

lim inf
r→+∞

logp+1 T (r,f)

logq ϕ (r)
≤ lim inf

r→+∞
r∈E3�E0

logp+1 T (r,f)

logq ϕ (r)
≤ µ[p,q] (As,ϕ) + 3ε.

Since ε > 0 is arbitrary, we have µ[p+1,q] (f,ϕ) ≤ µ[p,q] (As,ϕ) .

Lemma 2.9 Let p,q be integers such that p ≥ q > 1 or p > q = 1 and let
A0, · · · ,An−1 be meromorphic functions. Assume that λ[p,q]

(
1
A0
,ϕ
)
< µ[p,q] (A0,ϕ)

and that max
{
σ[p,q] (Aj,ϕ) : j = 1, · · · ,n− 1

}
≤ µ[p,q] (A0,ϕ) = µ, 0 < µ < ∞,

and max
{
τ[p,q] (Aj,ϕ) : σ[p,q] (Aj,ϕ) = µ[p,q] (A0,ϕ) ,j 6= 0

}
< τ [p,q] (A0,ϕ) = τ,

0 < τ < ∞, where ϕ satisfies the conditions (i) − (ii) of Remark 1.2. If f ( 6≡ 0) is
a meromorphic solution of (1.1) , then we have

µ[p+1,q] (f,ϕ) ≥ µ[p,q] (A0,ϕ) .



188 BOUABDELLI AND BELAÏDI

Proof. Suppose that f (6≡ 0) is a meromorphic solution of (1.1) . By (1.1) , we obtain

(2.14) −A0 =
f(n)

f
+ An−1

f(n−1)

f
+ · · · + A1

f′

f
.

By λ[p,q]

(
1
A0
,ϕ
)
< µ[p,q] (A0,ϕ) , we have N (r,A0) = o (T (r,A0)) , r → +∞.

Then by (2.14) , we get
(2.15)

T (r,A0) = m (r,A0) + N (r,A0) ≤
n−1∑
j=1

m (r,Aj) +

n−1∑
j=1

m

(
r,
f(j)

f

)
+ o (T (r,A0)) .

Hence, by (2.15) and Lemma 2.3 that

(2.16) T (r,A0) ≤ O


n−1∑
j=1

m (r,Aj) + log (rT (r,f))


 ,

for sufficiently large r → +∞, r /∈ E2, where E2 is a set of r of finite linear measure.
Set b = max{σ[p,q] (Aj,ϕ) : σ[p,q] (Aj,ϕ) < µ[p,q] (A0,ϕ) = µ, j = 1, · · · ,n− 1}. If
σ[p,q] (Aj,ϕ) < µ[p,q] (A0,ϕ) = µ, then for any ε (0 < 2ε < µ− b) and all r → +∞,
we have

m (r,Aj) ≤ T (r,Aj) ≤ expp
{

(b + ε) logq ϕ (r)
}

(2.17) < expp
{

(µ−ε) logq ϕ (r)
}

= expp−1

{(
logq−1 ϕ (r)

)µ−ε}
.

Set τ1 = max
{
τ[p,q] (Aj,ϕ) : σ[p,q] (Aj,ϕ) = µ[p,q] (A0,ϕ) , j 6= 0

}
, then τ1 < τ. If

σ[p,q] (Aj,ϕ) = µ[p,q] (A0,ϕ) , τ[p,q] (Aj,ϕ) ≤ τ1 < τ, then for r → +∞ and any
ε (0 < 2ε < τ − τ1) , we have

(2.18) m (r,Aj) ≤ T (r,Aj) < expp−1
{

(τ1 + ε)
(
logq−1 ϕ (r)

)µ}
.

By the definition of the lower [p,q] −ϕ type, for r → +∞, we have

(2.19) T (r,A0) > expp−1
{

(τ −ε)
(
logq−1 ϕ (r)

)µ}
.

When p ≥ q > 1 or p > q = 1, we have for r → +∞

expp−1
{

(τ1 + ε)
(
logq−1 ϕ (r)

)µ}
= o

(
expp−1

{
(τ −ε)

(
logq−1 ϕ (r)

)µ})
.

By substituting (2.17) − (2.19) into (2.16) , we obtain

(2.20) expp−1
{

(τ − 2ε)
(
logq−1 ϕ (r)

)µ} ≤ O (log (rT (r,f))) , r /∈ E2,r → +∞.
Then by (2.20) , Remark 1.2 and Lemma 2.2, we have µ[p+1,q] (f,ϕ) ≥ µ[p,q] (A0,ϕ) .

Lemma 2.10 Let p,q be integers such that p ≥ q ≥ 1 and let f be a meromorphic
function with 0 < σ[p,q] (f,ϕ) < ∞, where ϕ (r) only satisfies lim

r→+∞
logq−1 ϕ(αr)

logq−1 ϕ(r)
= 1

for some α > 1. Then for every ε > 0, there exists a set E5 ⊂ (1,∞) of infinite
logarithmic measure such that

τ[p,q] (f,ϕ) = lim
r→+∞
r∈E5

logp−1 T (r,f)(
logq−1 ϕ (r)

)σ[p,q](f,ϕ) .



GROWTH AND COMPLEX OSCILLATION OF LINEAR DIFFERENTIAL EQUATIONS 189

Proof. By the definition of the [p,q] − ϕ type, there exists a sequence {rn}
∞
n=1

tending to ∞ satisfying
(
1 + 1

n

)
rn < rn+1, and

τ[p,q] (f,ϕ) = lim
rn→∞

logp−1 T (rn,f)(
logq−1 ϕ (rn)

)σ[p,q](f,ϕ) .
Then for any given ε > 0, there exists an n1 such that for n ≥ n1 and any r ∈[
rn,
(
1 + 1

n

)
rn
]
, we have

logp−1 T (rn,f)(
logq−1 ϕ (rn)

)σ[p,q](f,ϕ)
(

logq−1 ϕ (rn)

logq−1 ϕ
[(

1 + 1
n

)
rn
])σ[p,q](f,ϕ)

≤
logp−1 T (r,f)(

logq−1 ϕ (r)
)σ[p,q](f,ϕ) .

When q ≥ 1, we have logq−1 ϕ(rn)
logq−1 ϕ[(1+

1
n )rn]

→ 1, rn →∞. Set

E5 =

∞⋃
n=n1

[
rn,

(
1 +

1

n

)
rn

]
.

Then, we have

lim
r→+∞
r∈E5

logp−1 T (r,f)(
logq−1 ϕ (r)

)σ[p,q](f,ϕ) ≥ limrn→∞ logp−1 T (rn,f)(logq−1 ϕ (rn))σ[p,q](f,ϕ) = τ[p,q] (f,ϕ)
and

∫
E5

dr
r

=
∞∑

n=n1

(1+ 1n )rn∫
rn

dt
t

=
∞∑

n=n1

log
(
1 + 1

n

)
= ∞. Therefore, by the evident

fact that

lim
r→+∞
r∈E5

logp−1 T (r,f)(
logq−1 ϕ (r)

)σ[p,q](f,ϕ) ≤ lim supr→+∞
r∈E5

logp−1 T (r,f)(
logq−1 ϕ (r)

)σ[p,q](f,ϕ) = τ[p,q] (f,ϕ) ,
we have

τ[p,q] (f,ϕ) = lim
r→+∞
r∈E5

logp−1 T (r,f)(
logq−1 ϕ (r)

)σ[p,q](f,ϕ) .
The proof of the following two lemmas is essentially the same as in the corre-

sponding results for the usual order and lower order. For details, see Chapter 2 of
the book by Goldberg-Ostrovskii [7] and Chapter 1 of the book by Yang-Yi [18].
So, we omit the proofs.

Lemma 2.11 Let p ≥ q ≥ 1 be integers, and let f and g be non-constant mero-
morphic functions of [p,q] −ϕ order. Then we have

ρ[p,q] (f + g,ϕ) ≤ max
{
ρ[p,q] (f,ϕ) ,ρ[p,q] (g,ϕ)

}
and

ρ[p,q] (fg,ϕ) ≤ max
{
ρ[p,q] (f,ϕ) ,ρ[p,q] (g,ϕ)

}
.

Furthermore, if ρ[p,q] (f,ϕ) > ρ[p,q] (g,ϕ) , then we obtain

ρ[p,q] (f + g,ϕ) = ρ[p,q] (fg,ϕ) = ρ[p,q] (f,ϕ) .



190 BOUABDELLI AND BELAÏDI

Lemma 2.12 Let p ≥ q ≥ 1 be integers, and let f and g be non-constant mero-
morphic functions with ρ[p,q] (f,ϕ) as [p,q]−ϕ order of f and µ[p,q] (g,ϕ) as lower
[p,q] −ϕ order of g. Then we have

µ[p,q] (f + g,ϕ) ≤ max
{
ρ[p,q] (f,ϕ) ,µ[p,q] (g,ϕ)

}
and

µ[p,q] (fg,ϕ) ≤ max
{
ρ[p,q] (f,ϕ) ,µ[p,q] (g,ϕ)

}
.

Furthermore, if µ[p,q] (g,ϕ) > ρ[p,q] (f,ϕ) , then we obtain

µ[p,q] (f + g,ϕ) = µ[p,q] (fg,ϕ) = µ[p,q] (g,ϕ) .

3. Proof of theorems

Proof of Theorem 1.1 By Lemma 1.1 and (2.10) , we have as in Lemma 2.8

T (r,f) ≤ expp+1
{(
σ[p,q] (A0,ϕ) + 3ε

)
logq ϕ (r)

}
,

for any ε > 0 and r /∈ E0,r → +∞, where E0 is a set of r of finite logarith-
mic measure. By Lemma 2.2, we get σ[p+1,q] (f,ϕ) ≤ σ[p,q] (A0,ϕ) . Set d =
max{σ[p,q] (Aj,ϕ) : σ[p,q] (Aj,ϕ) < σ[p,q] (A0,ϕ) ,j = 1, · · · ,n−1}. If σ[p,q] (Aj,ϕ) <
µ[p,q] (A0,ϕ) ≤ σ[p,q] (A0,ϕ) or σ[p,q] (Aj,ϕ) ≤ µ[p,q] (A0,ϕ) < σ[p,q] (A0,ϕ) , then
for any given ε

(
0 < 2ε < σ[p,q] (A0,ϕ) −d

)
and sufficiently large r, we have

(3.1) T (r,Aj) ≤ expp
{

(d + ε) logq ϕ (r)
}

= expp−1

{(
logq−1 ϕ (r)

)d+ε}
.

Set τ1 = max{τ[p,q] (Aj,ϕ) : σ[p,q] (Aj,ϕ) = µ[p,q] (A0,ϕ) ,j 6= 0}. If

σ[p,q] (Aj,ϕ) = µ[p,q] (A0,ϕ) = σ[p,q] (A0,ϕ) ,

then we have τ1 < τ ≤ τ[p,q] (A0,ϕ) . Therefore

(3.2) T (r,Aj) ≤ expp−1
{

(τ1 + ε)
(
logq−1 ϕ (r)

)σ[p,q](A0,ϕ)}
,

holds for any r → +∞ and any given ε
(
0 < 2ε < τ[p,q] (A0,ϕ) − τ1

)
. By the defi-

nition of the [p,q] −ϕ type and Lemma 2.10, and sufficiently large r ∈ E5, where
E5 is a set of r of infinite logarithmic measure, we have

(3.3) T (r,A0) > expp−1

{(
τ[p,q] (A0,ϕ) −ε

)(
logq−1 ϕ (r)

)σ[p,q](A0,ϕ)}
.

Then by (2.16) and (3.1) − (3.3) , for all sufficiently large r, r ∈ E5�E2 and the
above ε, we obtain

(3.4) expp−1

{(
τ[p,q] (A0,ϕ) − 2ε

)(
logq−1 ϕ (r)

)σ[p,q](A0,ϕ)} ≤ O (log rT (r,f)) ,
where E2 is a set of r of finite linear measure. Then, we have

σ[p+1,q] (f,ϕ) ≥ σ[p,q] (A0,ϕ) .

Thus, we have σ[p+1,q] (f,ϕ) = σ[p,q] (A0,ϕ) . By Lemmas 2.8 and 2.9, we have

µ[p+1,q] (f,ϕ) = µ[p,q] (A0,ϕ) . Now we need to prove λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ)
and λ̄[p+1,q] (f −ψ,ϕ) = σ[p+1,q] (f,ϕ) . Setting g = f − ψ, since σ[p+1,q] (ψ,ϕ) <
µ[p,q] (A0,ϕ) , then by Lemmas 2.11 and 2.12 we have σ[p+1,q] (g,ϕ) = σ[p+1,q] (f,ϕ) =
σ[p,q] (A0,ϕ) , and

µ[p+1,q] (g,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (A0,ϕ) , λ̄[p+1,q] (g,ϕ) = λ̄[p+1,q] (f −ψ,ϕ)



GROWTH AND COMPLEX OSCILLATION OF LINEAR DIFFERENTIAL EQUATIONS 191

and λ̄[p+1,q] (g,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) . By substituting f = g + ψ,f′ = g′ +
ψ′, · · · ,f(n) = g(n) + ψ(n) into (1.1) , we get

(3.5) g(n) + An−1g
(n−1) + · · · + A0g = −

[
ψ(n) + An−1ψ

(n−1) + · · · + A0ψ
]
.

If F = ψ(n)+An−1ψ
(n−1)+· · ·+A0ψ ≡ 0, then by Lemma 2.9, we have µ[p+1,q] (ψ,ϕ) ≥

µ[p,q] (A0,ϕ) , which is a contradiction. Hence F (z) 6≡ 0. Since F (z) 6≡ 0 and
σ[p+1,q] (F,ϕ) ≤ σ[p+1,q] (ψ,ϕ) < µ[p,q] (A0,ϕ)

= µ[p+1,q] (f,ϕ) = µ[p+1,q] (g,ϕ) ≤ σ[p+1,q] (g,ϕ) = σ[p+1,q] (f,ϕ) ,
then by Lemma 2.7 and (3.5) , we have λ̄[p+1,q] (g,ϕ) = λ[p+1,q] (g,ϕ) = σ[p+1,q] (g,ϕ) =

σ[p,q] (A0,ϕ) , i.e., λ̄[p+1,q] (f −ψ,ϕ) = λ[p+1,q] (f −ψ,ϕ) = σ[p+1,q] (f,ϕ) = σ[p,q] (A0,ϕ) .
By Lemma 2.6 and (3.5) , we have λ̄[p+1,q] (g,ϕ) = µ[p+1,q] (g,ϕ) , i.e.,

λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (A0,ϕ) .
Therefore

λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (A0,ϕ)
≤ σ[p,q] (A0,ϕ) = σ[p+1,q] (f,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) = λ[p+1,q] (f −ψ,ϕ) .

The proof of the theorem is complete.

Proof of Theorem 1.2 By the first part of the proof of Theorem 1.1, we can get
σ[p+1,q] (f,ϕ) ≤ σ[p,q] (A0,ϕ) . By

(3.6) lim sup
r→+∞

n−1∑
j=1

m (r,Aj)

m (r,A0)
< 1

we have for r → +∞

(3.7)

n−1∑
j=1

m (r,Aj) < δm (r,A0) ,

where δ ∈ (0, 1) . By λ[p,q]
(

1
A0
,ϕ
)
< µ[p,q] (A0,ϕ) , we have N (r,A0) = o (T (r,A0)) ,

r → +∞. By (2.15) and (3.7), for r → +∞,r /∈ E2, we obtain
(3.8)
T (r,A0) = m (r,A0) + N (r,A0) ≤ δT (r,A0) + O (log rT (r,f)) + o (T (r,A0)) ,

where E2 is a set of r of finite linear measure. By Lemma 2.2 and (3.8) , we have
σ[p+1,q] (f,ϕ) ≥ σ[p,q] (A0,ϕ) . Then we have σ[p+1,q] (f,ϕ) = σ[p,q] (A0,ϕ) . By (3.8)
and Lemma 2.2, we have µ[p+1,q] (f,ϕ) ≥ µ[p,q] (A0,ϕ) . By Lemma 2.8, we have
µ[p+1,q] (f,ϕ) ≤ µ[p,q] (A0,ϕ) , then we get

µ[p+1,q] (f,ϕ) = µ[p,q] (A0,ϕ) .

By using the similar proof of Theorem 1.1, we can get

λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (A0,ϕ) ≤ σ[p,q] (A0,ϕ)
= σ[p+1,q] (f,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) = λ[p+1,q] (f −ψ,ϕ) .

The proof of the theorem is complete.

Proof of Theorem 1.3 Suppose that f is rational solution of (1.1) . If f is either a
rational function with a pole of multiplicity n ≥ 1 at z0 or a polynomial with degree



192 BOUABDELLI AND BELAÏDI

deg (f) ≥ s, then f(s) (z) 6≡ 0. If max{σ[p,q] (Aj,ϕ) ,j 6= s} < µ[p,q] (As,ϕ) = µ,
then we have

µ[p,q] (0,ϕ) = µ[p,q]

(
f(n) + An−1f

(n−1) + · · · + A0f,ϕ
)

= µ[p,q] (As,ϕ) = µ > 0,

which is a contradiction. Set

τ1 = max{τ[p,q] (Aj,ϕ) : σ[p,q] (Aj,ϕ) = µ[p,q] (As,ϕ) ,j 6= s}.

If σ[p,q] (Aj,ϕ) = µ[p,q] (As,ϕ) , τ[p,q] (Aj,ϕ) ≤ τ1 < τ, then we may choose con-
stants δ1,δ2 such that τ1 < δ1 < δ2 < τ. For sufficiently large r, we have

(3.9) m (r,Aj) ≤ T (r,Aj) ≤ expp−1
{
δ1
(
logq−1 ϕ (r)

)µ}
.

If σ[p,q] (Aj,ϕ) < µ[p,q] (As,ϕ) , then for sufficiently large r

and any given ε
(
0 < 2ε < µ[p,q] (As,ϕ) −σ[p,q] (Aj,ϕ)

)
, we obtain

(3.10) m (r,Aj) ≤ T (r,Aj) ≤ expp
{(
σ[p,q] (Aj,ϕ) + ε

)
logq ϕ (r)

}
.

Under the assumption that λ[p,q]

(
1
As
,ϕ
)
< µ[p,q] (As,ϕ) , for sufficiently large r,

we have

(3.11) N (r,As) = o (T (r,As)) .

By the definition of the lower [p,q] −ϕ type, for sufficiently large r, we get

(3.12) T (r,As) ≥ expp−1
{
δ2
(
logq−1 ϕ (r)

)µ}
.

By (1.1) , we have

(3.13) T (r,As) ≤ N (r,As) +
∑
j 6=s

m (r,Aj) + O (log r) ,

for sufficiently large r. Hence, by substituting (3.9) , (3.10) and (3.11) into (3.13)
we have the contradiction. Therefore, if f is a non-transcendental meromorphic
solution, then it must be a polynomial with degree deg (f) ≤ s− 1.

Now, we assume that f is a transcendental meromorphic solution of (1.1) . By
(1.1) , we have

(3.14) −As =
f

f(s)

[
f(n)

f
+ · · · + As+1

f(s+1)

f
+ As−1

f(s−1)

f
+ · · · + A0

]
.

Noting that

m

(
r,

f

f(s)

)
≤ T (r,f) + T

(
r,

1

f(s)

)
= T (r,f) + T

(
r,f(s)

)
+ O (1)

≤ T (r,f) + (s + 1) T (r,f) + o (T (r,f)) + O (1)

(3.15) = (s + 2) T (r,f) + o (T (r,f)) + O (1) .

By Lemma 2.3, (3.14) and (3.15) , we obtain

T (r,As) = m (r,As) + N (r,As)

(3.16) ≤ N (r,As) +
∑
j 6=s

m (r,Aj) + (s + 3) T (r,f) + O (log (rT (r,f))) ,



GROWTH AND COMPLEX OSCILLATION OF LINEAR DIFFERENTIAL EQUATIONS 193

for sufficiently large r /∈ E2, where E2 is a set of r of finite linear measure. Then
by (3.9)−(3.12) , (3.16) and Lemma 2.2, we can get µ[p,q] (f,ϕ) ≥ µ[p,q] (As,ϕ) and
σ[p,q] (f,ϕ) ≥ σ[p,q] (As,ϕ) . By Lemma 1.1 and (2.10) , we have

(3.17) T (r,f) ≤ expp+1
{(
σ[p,q] (As,ϕ) + 3ε

)
logq ϕ (r)

}
,

for any ε > 0, and r /∈ E0, r → +∞, where E0 is a set of r of linear logarithmic
measure. Then by (3.17) and Lemma 2.2, we have σ[p+1,q] (f,ϕ) ≤ σ[p,q] (As,ϕ) . By
Lemma 2.8, we obtain µ[p+1,q] (f,ϕ) ≤ µ[p,q] (As,ϕ) . Then we get σ[p+1,q] (f,ϕ) ≤
σ[p,q] (As,ϕ) ≤ σ[p,q] (f,ϕ) and

µ[p+1,q] (f,ϕ) ≤ µ[p,q] (As,ϕ) ≤ µ[p,q] (f,ϕ) .
The proof of the theorem is complete.

Acknowledgements. This paper is supported by University of Mostaganem (UM-
AB) (CNEPRU Project Code B02220120024).

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194 BOUABDELLI AND BELAÏDI

Department of Mathematics, Laboratory of Pure and Applied Mathematics, Univer-

sity of Mostaganem (UMAB), B. P. 227 Mostaganem, Algeria

∗Corresponding author