International Journal of Analysis and Applications
ISSN 2291-8639
Volume 4, Number 2 (2014), 159-173
http://www.etamaths.com

PROPERTIES OF SOLUTIONS OF COMPLEX DIFFERENTIAL

EQUATIONS IN THE UNIT DISC

ZINELÂABIDINE LATREUCH AND BENHARRAT BELAÏDI∗

Abstract. In this paper, we investigate the growth and oscillation of higher
order differential polynomial with meromorphic coefficients in the unit disc

∆ = {z : |z| < 1} generated by solutions of the linear differential equation

f(k) + A (z) f = 0 (k ≥ 2) ,
where A (z) is a meromorphic function of finite iterated p−order in ∆.

1. Introduction and main results

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 on
the complex plane and in the unit disc ∆ = {z : |z| < 1} (see [14] , [15] , [18] , [20] , [22]).
We need to give some definitions and discussions. Firstly, let us give two defini-
tions about the degree of small growth order of functions in ∆ as polynomials on
the complex plane C. There are many types of definitions of small growth order of
functions in ∆ (see [11] , [12]) .

Definition 1.1 Let f be a meromorphic function in ∆, and

D (f) := lim sup
r→1−

T (r,f)

log 1
1−r

= b.

If b < ∞, we say that f is of finite b degree (or is non-admissible). If b = ∞, we say
that f is of infinite degree (or is admissible), both defined by characteristic function
T(r,f).

Definition 1.2 Let f be an analytic function in ∆, and

DM (f) := lim sup
r→1−

log+ M (r,f)

log 1
1−r

= a < ∞ (or a = ∞) ,

then we say that f is a function of finite a degree (or of infinite degree) defined by
maximum modulus function M(r,f) = max

|z|=r
|f (z)| .

Now we give the definitions of iterated order and growth index to classify generally
the functions of fast growth in ∆ as those in C (see [5] , [17] , [18]). Let us define

2010 Mathematics Subject Classification. 34M10, 30D35.
Key words and phrases. Iterated p−order, Linear differential equations, Iterated exponent of

convergence of the sequence of distinct zeros, Unit disc, Differential polynomials.

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.

159



160 LATREUCH AND BELAÏDI

inductively, for r ∈ [0, 1) , exp1 r = er and expp+1 r = exp
(
expp r

)
, p ∈ N. We also

define for all r sufficiently large in (0, 1) , log1 r = log r and logp+1 r = log
(
logp r

)
,

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

Definition 1.3 [6] The iterated p−order of a meromorphic function f in ∆ is
defined as

ρp (f) = lim sup
r→1−

log+p T (r,f)

log 1
1−r

(p ≥ 1) .

For an analytic function f in ∆, we also define

ρM,p (f) = lim sup
r→1−

log+p+1 M (r,f)

log 1
1−r

(p ≥ 1) .

Remark 1.1 It follows by M. Tsuji in [22] that if f is an analytic function in ∆,
then

ρ1 (f) ≤ ρM,1 (f) ≤ ρ1 (f) + 1.
However, it follows by Proposition 2.2.2 in [18]

ρM,p (f) = ρp (f) (p ≥ 2) .
Definition 1.4 [6] The growth index of the iterated order of a meromorphic func-
tion f(z) in ∆ is defined as

i (f) =




0, if f is non-admissible,
min{j ∈ N : ρj (f) < ∞} , if f is admissible

and ρj (f) < ∞ for some j ∈ N,
+∞, if ρj (f) = ∞ for all j ∈ N.

For an analytic function f in ∆, we also define

iM (f) =




0, if f is of finite degree,
min{j ∈ N : ρM,j (f) < ∞} , if f is of infinite degree

and ρM,j (f) < ∞ for some j ∈ N,
+∞, if ρM,j (f) = ∞ for all j ∈ N.

Definition 1.5 ([13] , [16]) The iterated p−type of a meromorphic function f of
iterated p−order ρp (f) (0 < ρp (f) < ∞) in ∆ is defined as

τp (f) = lim sup
r→1−

(1 −r)ρp(f) log+p−1 T (r,f) .

Definition 1.6 [7] Let f be a meromorphic function in ∆. Then the iterated
exponent of convergence of the sequence of zeros of f (z) is defined as

λp (f) = lim sup
r→1−

log+p N
(
r, 1
f

)
log 1

1−r
,

where N
(
r, 1
f

)
is the counting function of zeros of f (z) in {z ∈ C : |z| < r} . Simi-

larly, the iterated exponent of convergence of the sequence of distinct zeros of f (z)
is defined as

λp (f) = lim sup
r→1−

log+p N
(
r, 1
f

)
log 1

1−r
,



PROPERTIES OF SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS 161

where N
(
r, 1
f

)
is the counting function of distinct zeros of f (z) in {z ∈ C : |z| < r}.

Definition 1.7 [7] The growth index of the convergence exponent of the sequence
of zeros of a meromorphic f(z) in ∆ is defined as

iλ (f) =




0, if N
(
r, 1
f

)
= O

(
log 1

1−r

)
,

min{j ∈ N : λj (f) < ∞} , if some j ∈ N with λj (f) < ∞ exists,
+∞, if λj (f) = ∞ for all j ∈ N.

Remark 1.2 Similarly, we can define the finiteness degree iλ (f) of λp(f).

Consider for k ≥ 2 the complex differential equation

(1.1) f(k) + A (z) f = 0

and the differential polynomial

(1.2) gf = dkf
(k) + dk−1f

(k−1) + · · · + d1f′ + d0f,
where A and dj (j = 0, 1, · · · ,k) are meromorphic functions in ∆.

Let L(G) denote a differential subfield of the field M(G) of meromorphic
functions in a domain G ⊂ C. Throughout this paper, we simply denote L instead
of L(∆) . Special case of such differential subfield

Lp+1,ρ={g meromorphic in ∆: ρp+1 (g) < ρ} ,
where ρ is a positive constant. Recently, T. B. Cao, H. Y. Xu and C. X. Zhu [8], T.
B. Cao, L. M. Li, J. Tu and H. Y. Xu [10] have studied the complex oscillation of
differential polynomial generated by meromorphic and analytic solutions of second
order linear differential equations with meromorphic coefficients and obtained the
following results.

Theorem A [10] Let A (z) be an analytic function of infinite degree and of finite
iterated order ρM,p (A) = ρ > 0 in the unit disc ∆, and let f 6≡ 0 be a solution of
the equation

(1.3) f′′ + A (z) f = 0.

Moreover, let

(1.4) P [f] = P
(
f,f′, · · · ,f(m)

)
=

m∑
j=0

pjf
(j)

be a linear differential polynomial with analytic coefficients pj ∈ Lp+1,ρ, assuming
that at least one of the coefficients pj does vanish identically. If ϕ (z) ∈ Lp+1,ρ
is a non-zero analytic function in ∆, and neither P [f] nor P [f] − ϕ vanishes
identically, then we have

iλ (P [f] −ϕ) = i (f) = p + 1
and

λp+1 (P [f] −ϕ) = ρM,p+1 (f) = ρM,p (A) = ρ.
Theorem B [8] Let A be an admissible meromorphic function of finite iterat-
ed order ρp (A) = ρ > 0 (1 ≤ p < ∞) in the unit disc ∆ such that δ (∞,A) =



162 LATREUCH AND BELAÏDI

lim inf
r→1−

m(r,A)
T(r,A)

= δ > 0, and let f be a non-zero meromorphic solution of equation

(1.3) such that δ (∞,f) > 0. Moreover, let be a linear differential polynomial (1.4)
with meromorphic coefficients pj ∈ Lp+1,ρ, assuming that at least one of the coef-
ficients pj does not vanish identically. If ϕ ∈ Lp+1,ρ is a non-zero meromorphic
function in ∆, and neither P [f] nor P [f] −ϕ vanishes identically, then we have

i (f) = iλ (P [f] −ϕ) = p + 1
and

λp (P [f] −ϕ) = ρp+1 (f) = ρp (A) = ρ
if p > 1, while

ρp (A) ≤ λp (P [f] −ϕ) ≤ ρp+1 (f) ≤ ρp (A) + 1
if p = 1.

Remark 1.3 The idea of the proofs of Theorems A-B is borrowed from the paper
of Laine, Rieppo [19] with the modifications reflecting the change from the complex
plane C to the unit disc ∆.

Before we state our results, we define the sequence of meromorphic functions αi,j
(j = 0, · · · ,k − 1) in ∆ by

(1.5) αi,j =

{
α′i,j−1 + αi−1,j−1, for all i = 1, · · · ,k − 1,

α′0,j−1 −Aαk−1,j−1, for i = 0
and

(1.6) αi,0 =

{
di, for all i = 1, · · · ,k − 1,

d0 −dkA, for i = 0.
We define also h and ψ (z) by

(1.7) h =

∣∣∣∣∣∣∣∣∣∣
α0,0 α1,0 . . αk−1,0
α0,1 α1,1 . . αk−1,1
. . . . .
. . . . .

α0,k−1 α1,k−1 . . αk−1,k−1

∣∣∣∣∣∣∣∣∣∣
,

(1.8) ψ (z) = C0ϕ + C1ϕ
′ + · · · + Ck−1ϕ(k−1),

where Cj (j = 0, · · · ,k − 1) are finite iterated p−order meromorphic functions in
∆ depending on αi,j and ϕ 6≡ 0 is a meromorphic function in ∆ with ρp (ϕ) < ∞.

The main purpose of this paper is to study the growth and oscillation of differential
polynomial (1.2) generated by meromorphic solutions of equation (1.1) in the unit
disc ∆.

Theorem 1.1 Suppose that A (z) is a meromorphic function of finite iterated
p−order in ∆ and that dj (z) (j = 0, 1, · · · ,k) are finite iterated p−order meromor-
phic functions in ∆ that are not all vanishing identically such that h 6≡ 0. If f (z)
is an infinite iterated p−order meromorphic solution of (1.1) with ρp+1 (f) = ρ,
then the differential polynomial (1.2) satisfies

ρp (gf ) = ρp (f) = ∞



PROPERTIES OF SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS 163

and
ρp+1 (gf ) = ρp+1 (f) = ρ.

Furthermore, if f is a finite iterated p−order meromorphic solution of (1.1) such
that

(1.9) ρp (f) > max{ρp (A) ,ρp (dj) (j = 0, 1, · · · ,k)} ,
then

ρp (gf ) = ρp (f) .

Remark 1.4 In Theorem 1.1, if we do not have the condition h 6≡ 0, then the
conclusions of Theorem 1.1 cannot hold. For example, if we take dk = 1,d0 = A
and dj ≡ 0 (j = 1, · · · ,k − 1) , then h ≡ 0. It follows that gf ≡ 0 and ρp (gf ) = 0.
So, if f (z) is an infinite iterated p−order meromorphic solution of (1.1) , then
ρp (gf ) = 0 < ρp (f) = ∞, and if f is a finite iterated p−order meromorphic
solution of (1.1) such that (1.9) holds, then ρp (gf ) = 0 < ρp (f).

Corollary 1.1 Suppose that A (z) is admissible meromorphic function in ∆ such
that i (A) = p (1 ≤ p < ∞) and δ (∞,A) = δ > 0. Let dj (z) (j = 0, 1, · · · ,k)
be finite iterated p−order meromorphic functions in ∆ that are not all vanishing
identically such that h 6≡ 0, and let f be a nonzero meromorphic solution of (1.1) .
If δ (∞,f) > 0, then the differential polynomial gf satisfies i (gf ) = p + 1 and
ρp+1 (gf ) = ρp+1 (f) = ρp (A) if p > 1, while

ρp (A) ≤ ρp+1 (gf ) = ρp+1 (f) ≤ ρp (A) + 1
if p = 1.

Theorem 1.2 Under the assumptions of Theorem 1.1, let ϕ (z) 6≡ 0 be a meromor-
phic function with finite iterated p−order in ∆ such that ψ (z) is not a solution of
(1.1) . If f (z) is an infinite iterated p−order meromorphic solution of (1.1) with
ρp+1 (f) = ρ, then the differential polynomial (1.2) satisfies

λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) = ∞
and

λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) = ρ.
Furthermore, if f is a finite iterated p−order meromorphic solution of (1.1) such
that

(1.10) ρp (f) > max{ρp (A) ,ρp (ϕ) ,ρp (dj) (j = 0, 1, · · · ,k)} ,
then

λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) .
Corollary 1.2 Under the assumptions of Corollary 1.1, let ϕ (z) 6≡ 0 be a mero-
morphic function with finite iterated p−order in ∆ such that ψ (z) 6≡ 0. Then the
differential polynomial (1.2) satisfies

λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) = ρp (A)
if p > 1, while

ρp (A) ≤ λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) ≤ ρp (A) + 1
if p = 1.



164 LATREUCH AND BELAÏDI

Remark 1.5 The ideas of the proofs of Theorems 1.1 and 1.2 are from [21] with
modification from the complex plane C to the unit disc ∆. For some papers related
in the complex plane see [19, 21, 4] .

2. Auxiliary lemmas

Lemma 2.1 [9] Let A0, A1, · · · , Ak−1, F 6≡ 0 be meromorphic functions in ∆, and
let f be a meromorphic solution of the equation

(2.1) f(k) + Ak−1 (z) f
(k−1) + · · · + A1 (z) f′ + A0 (z) f = F (z)

such that

max{ρp (Aj) (j = 0, 1, · · · ,k − 1) ,ρp (F)} < ρp (f) ≤ +∞.

Then

λp (f) = λp (f) = ρp (f)

and

λp+1 (f) = λp+1 (f) = ρp+1 (f) .

Lemma 2.2 [6] Let p ≥ 1 be an integer, and let A0(z), · · · ,Ak−1(z) be analytic
functions in ∆ such that i (A0) = p. If

max{i (Aj) : j = 1, · · · ,k − 1} < p

or

max{ρp (Aj) : j = 1, · · · ,k − 1} < ρp (A0) ,
then every solution f 6≡ 0 of the equation

(2.2) f(k) + Ak−1 (z) f
(k−1) + · · · + A1 (z) f′ + A0 (z) f = 0,

satisfies i (f) = p + 1 and ρp (f) = ∞, ρp (A0) ≤ ρp+1 (f) = ρM,p+1 (f) ≤
max{ρM,p (Aj) : j = 0, 1, · · · ,k − 1}.

Lemma 2.3 [3] Let f and g be meromorphic functions in the unit disc ∆ such that
0 < ρp (f) ,ρp (g) < ∞ and 0 < τp (f) ,τp (g) < ∞. Then we have
(i) If ρp (f) > ρp (g) , then we obtain

(2.3) τp (f + g) = τp (fg) = τp (f) .

(ii) If ρp (f) = ρp (g) and τp (f) 6= τp (g) , then we get

(2.4) ρp (f + g) = ρp (fg) = ρp (f) = ρp (g) .

Lemma 2.4 [17, 2] Let f be a meromorphic function in the unit disc for which
i (f) = p ≥ 1 and ρp (f) = β < ∞, and let k ∈ N. Then for any ε > 0,

(2.5) m

(
r,
f(k)

f

)
= O

(
expp−2

(
log

1

1 −r

)β+ε)
for all r outside a set E ⊂ [0, 1) with

∫
E

dr
1−r < ∞.

Lemma 2.5 [8] Let A (z) be an admissible meromorphic function in ∆ such that
i (A) = p (1 ≤ p < ∞) and δ (∞,A) = δ > 0, and let f be a nonzero meromorphic



PROPERTIES OF SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS 165

solution of (1.1) . If δ (∞,f) > 0, then i (f) = p + 1 and ρp+1 (f) = ρp (A) if
p > 1, while

ρp (A) ≤ ρp+1 (f) ≤ ρp (A) + 1
if p = 1.

Lemma 2.6 [1] Let g : (0, 1) → R and h : (0, 1) → R be monotone increasing
functions such that g (r) ≤ h (r) holds outside of an exceptional set E1 ⊂ [0, 1)
for which

∫
E1

dr
1−r < ∞. Then there exists a constant d ∈ (0, 1) such that if

s (r) = 1 −d (1 −r) , then g (r) ≤ h (s (r)) for all r ∈ [0, 1).

3. Proofs of the Theorems and the Corollaries

Proof of Theorem 1.1 Suppose that f is an infinite iterated p−order meromor-
phic solution of (1.1) with ρp+1 (f) = ρ. By (1.1) we have

(3.1) f(k) = −Af

which implies

gf = dkf
(k) + dk−1f

(k−1) + · · · + d0f

(3.2) = dk−1f
(k−1) + · · · + (d0 −dkA) f.

We can rewrite (3.2) as

(3.3) gf =

k−1∑
i=0

αi,0f
(i),

where αi,0 are defined in (1.6) . Differentiating both sides of equation (3.3) and

replacing f(k) with f(k) = −Af, we obtain

g′f =

k−1∑
i=0

α′i,0f
(i) +

k−1∑
i=0

αi,0f
(i+1) =

k−1∑
i=0

α′i,0f
(i) +

k∑
i=1

αi−1,0f
(i)

= α′0,0f +

k−1∑
i=1

α′i,0f
(i) +

k−1∑
i=1

αi−1,0f
(i) + αk−1,0f

(k)

= α′0,0f +

k−1∑
i=1

(
α′i,0 + αi−1,0

)
f(i) −αk−1,0Af

(3.4) =

k−1∑
i=1

(
α′i,0 + αi−1,0

)
f(i) +

(
α′0,0 −αk−1,0A

)
f.

We can rewrite (3.4) as

(3.5) g′f =

k−1∑
i=0

αi,1f
(i),

where

(3.6) αi,1 =

{
α′i,0 + αi−1,0, for all i = 1, · · · ,k − 1,

α′0,0 −Aαk−1,0, for i = 0.



166 LATREUCH AND BELAÏDI

Differentiating both sides of equation (3.5) and replacing f(k) with f(k) = −Af, we
obtain

g′′f =

k−1∑
i=0

α′i,1f
(i) +

k−1∑
i=0

αi,1f
(i+1) =

k−1∑
i=0

α′i,1f
(i) +

k∑
i=1

αi−1,1f
(i)

= α′0,1f +

k−1∑
i=1

α′i,1f
(i) +

k−1∑
i=1

αi−1,1f
(i) + αk−1,1f

(k)

= α′0,1f +

k−1∑
i=1

(
α′i,1 + αi−1,1

)
f(i) −αk−1,1Af

(3.7) =

k−1∑
i=1

(
α′i,1 + αi−1,1

)
f(i) +

(
α′0,1 −αk−1,1A

)
f

which implies that

(3.8) g′′f =

k−1∑
i=0

αi,2f
(i),

where

(3.9) αi,2 =

{
α′i,1 + αi−1,1, for all i = 1, · · · ,k − 1,

α′0,1 −Aαk−1,1, for i = 0.
By using the same method as above we can easily deduce that

(3.10) g
(j)
f =

k−1∑
i=0

αi,jf
(i), j = 0, 1, · · · ,k − 1,

where

(3.11) αi,j =

{
α′i,j−1 + αi−1,j−1, for all i = 1, · · · ,k − 1,

α′0,j−1 −Aαk−1,j−1, for i = 0
and

(3.12) αi,0 =

{
di, for all i = 1, · · · ,k − 1,

d0 −dkA, for i = 0.
By (3.3) − (3.12) we obtain the system of equations

(3.13)




gf = α0,0f + α1,0f
′ + · · · + αk−1,0f(k−1),

g′f = α0,1f + α1,1f
′ + · · · + αk−1,1f(k−1),

g′′f = α0,2f + α1,2f
′ + · · · + αk−1,2f(k−1),
· · ·

g
(k−1)
f = α0,k−1f + α1,k−1f

′ + · · · + αk−1,k−1f(k−1).

By Cramer’s rule, and since h 6≡ 0 we have

(3.14) f =

∣∣∣∣∣∣∣∣∣∣

gf α1,0 . . αk−1,0
g′f α1,1 . . αk−1,1
. . . . .
. . . . .

g
(k−1)
f α1,k−1 . . αk−1,k−1

∣∣∣∣∣∣∣∣∣∣
h

.



PROPERTIES OF SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS 167

So, we obtain

(3.15) f = C0gf + C1g
′
f + · · · + Ck−1g

(k−1)
f ,

where Cj are finite iterated p−order meromorphic functions in ∆ depending on
αi,j, where αi,j are defined in (3.11) and (3.12) .

If ρp (gf ) < +∞, then by (3.15) we obtain ρp (f) < +∞, and this is a contra-
diction. Hence ρp (gf ) = ρp (f) = +∞.

Now, we prove that ρp+1 (gf ) = ρp+1 (f) = ρ. By (3.2), we get ρp+1 (gf ) ≤
ρp+1 (f) and by (3.15) we have ρp+1 (f) ≤ ρp+1 (gf ). This yield ρp+1 (gf ) =
ρp+1 (f) = ρ.

Furthermore, if f is a finite iterated p−order meromorphic solution of equation
(1.1) such that

(3.16) ρp (f) > max{ρp (A) ,ρp (dj) (j = 0, 1, · · · ,k)} ,

then

(3.17) ρp (f) > max{ρp (αi,j) : i = 0, · · · ,k − 1,j = 0, · · · ,k − 1} .

By (3.2) and (3.16) we have ρp (gf ) ≤ ρp (f) . Now, we prove ρp (gf ) = ρp (f) . If
ρp (gf ) < ρp (f) , then by (3.15) and (3.17) we get

ρp (f) ≤ max{ρp (Cj) (j = 0, · · · ,k − 1) ,ρp (gf )} < ρp (f)

and this is a contradiction. Hence ρp (gf ) = ρp (f) .

Remark 3.1 From (3.15) , it follows that the condition h 6≡ 0 is equivalent to that
gf,g

′
f,g
′′
f , ...,g

(k−1)
f are linearly independent over the field of meromorphic functions

of finite iterated p−order in ∆.

Proof of Corollary 1.1 Suppose f 6≡ 0 is a meromorphic solution of (1.1) . Then,
by Lemma 2.5, we have i (f) = p + 1 and ρp+1 (f) = ρp (A) if p > 1, while

ρp (A) ≤ ρp+1 (f) ≤ ρp (A) + 1

if p = 1. Thus, by Theorem 1.1 we obtain that the differential polynomial gf satisfies
i (gf ) = p + 1 and ρp+1 (gf ) = ρp+1 (f) = ρp (A) if p > 1, while

ρp (A) ≤ ρp+1 (gf ) = ρp+1 (f) ≤ ρp (A) + 1

if p = 1.

Proof of Theorem 1.2 Suppose that f is an infinite iterated p−order meromorphic
solution of equation (1.1) with ρp+1 (f) = ρ. Set w (z) = gf −ϕ. Since ρp (ϕ) < ∞,
then by Theorem 1.1 we have ρp (w) = ρp (gf ) = ∞ and ρp+1 (w) = ρp+1 (gf ) = ρ.
To prove λp (gf −ϕ) = λp (gf −ϕ) = ∞ and λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρ we
need to prove λp (w) = λp (w) = ∞ and λp+1 (w) = λp+1 (w) = ρ. By gf = w + ϕ
and (3.15) , we get

(3.18) f = C0w + C1w
′ + · · · + Ck−1w(k−1) + ψ (z) ,

where

(3.19) ψ (z) = C0ϕ + C1ϕ
′ + · · · + Ck−1ϕ(k−1).



168 LATREUCH AND BELAÏDI

Substituting (3.18) into (1.1) , we obtain

(3.20) Ck−1w
(2k−1) +

2k−2∑
i=0

φiw
(i) = −

(
ψ(k) + A (z) ψ

)
= H,

where φi (i = 0, · · · , 2k − 2) are meromorphic functions in ∆ with finite iterated
p−order. Since ψ (z) is not a solution of (1.1) , it follows that H 6≡ 0. Then by
Lemma 2.1, we obtain λp (w) = λp (w) = ∞ and λp+1 (w) = λp+1 (w) = ρ, i. e.,
λp (gf −ϕ) = λp (gf −ϕ) = ∞ and λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρ.

Suppose that f is a finite iterated p−order meromorphic solution of equation
(1.1) such that (1.10) holds. Set w (z) = gf −ϕ. Since ρp (ϕ) < ρp (f) , then by The-
orem 1.1 we have ρp (w) = ρp (gf ) = ρp (f) . To prove λp (gf −ϕ) = λp (gf −ϕ) =
ρp (f) we need to prove λp (w) = λp (w) = ρp (f) . Using the same reasoning as
above, we get

Ck−1w
(2k−1) +

2k−2∑
i=0

φiw
(i) = −

(
ψ(k) + A (z) ψ

)
= F,

where Ck−1, φi (i = 0, · · · , 2k − 2) are meromorphic functions in ∆ with finite it-
erated p−order ρp (Ck−1) < ρp (w) , ρp (φi) < ρp (w) (i = 0, · · · , 2k − 2) and

ψ (z) = C0ϕ + C1ϕ
′ + · · · + Ck−1ϕ(k−1), ρp (F) < ρp (w) .

Since ψ (z) is not a solution of (1.1) , it follows that F 6≡ 0. Then by Lemma 2.1,
we obtain λp (w) = λp (w) = ρp (f) , i. e., λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) .

Proof of Corollary 1.2 Suppose that f 6≡ 0 is a meromorphic solution of (1.1) .
Then, by Lemma 2.5, we have i (f) = p + 1 and ρp+1 (f) = ρp (A) if p > 1, while

ρp (A) ≤ ρp+1 (f) ≤ ρp (A) + 1

if p = 1. Since ψ 6≡ 0 and ρp (ψ) < ∞, then ψ cannot be a solution of equation
(1.1) . Thus, by Theorem 1.2 we obtain that the differential polynomial gf satisfies

λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) = ρp (A)

if p > 1, while

ρp (A) ≤ λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) ≤ ρp (A) + 1

if p = 1.

4. Discussions and applications

In this section, we consider the differential equation

(4.1) f′′′ + A (z) f = 0,

where A (z) is a meromorphic function of finite iterated p−order in ∆. It is clear
that the difficulty of the study of the differential polynomial generated by solutions
lies in the calculation of the coefficients αi,j. We explain here that by using our
method, the calculation of the coefficients αi,j can be deduced easily. We study for
example the growth of the differential polynomial

(4.2) gf = f
′′′ + f′′ + f′ + f.



PROPERTIES OF SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS 169

We have

(4.3)




gf = α0,0f + α1,0f
′ + α2,0f

′′,
g′f = α0,1f + α1,1f

′ + α2,1f
′′,

g′′f = α0,2f + α1,2f
′ + α2,2f

′′.

By (1.6) we obtain

(4.4) αi,0 =

{
1, for all i = 1, 2,
1 −A, for i = 0.

Now, by (3.6) we get

αi,1 =

{
α′i,0 + αi−1,0, for all i = 1, 2
α′0,0 −Aα2,0, for i = 0.

Hence

(4.5)




α0,1 = α
′
0,0 −Aα2,0 = −A′ −A,

α1,1 = α
′
1,0 + α0,0 = 1 −A,

α2,1 = α
′
2,0 + α1,0 = 1.

Finally, by (3.11) we have

αi,2 =

{
α′i,1 + αi−1,1, for all i = 1, 2,
α′0,1 −Aα2,1, for i = 0.

So, we obtain

(4.6)




α0,2 = α
′
0,1 −Aα2,1 = −A′′ −A′ −A,

α1,2 = α
′
1,1 + α0,1 = −2A′ −A,

α2,2 = α
′
2,1 + α1,1 = 1 −A.

Hence

(4.7)




gf = (1 −A) f + f′ + f′′,
g′f = (−A

′ −A) f + (1 −A) f′ + f′′,
g′′f = (−A

′′ −A′ −A) f + (−2A′ −A) f′ + (1 −A) f′′

and

h =

∣∣∣∣∣∣
1 −A 1 1
−A′ −A 1 −A 1
−A′′ −A′ −A −2A′ −A 1 −A

∣∣∣∣∣∣
(4.8) = 3A′ −A−AA′ −AA′′ + A2 −A3 + 2(A′)2 + 1.

Suppose that h 6≡ 0, by simple calculations we have

(4.9) f =
Ag′′f + (−1 − 2A

′) g′f +
(
1 −A + 2A′ + A2

)
gf

h

and by different conditions on the solution f we can ensure that

ρp (f
′′′ + f′′ + f′ + f) = ρp (f) .

Turning now to the problem of oscillation, for that we consider a meromorphic
function ϕ (z) 6≡ 0 of finite iterated p−order in ∆. From (4.9) we get

(4.10) f =
Aw′′ + (−1 − 2A′) w′ +

(
1 −A + 2A′ + A2

)
w

h
+ ψ (z) ,



170 LATREUCH AND BELAÏDI

where w = gf −ϕ and

(4.11) ψ (z) =
Aϕ′′ + (−1 − 2A′) ϕ′ +

(
1 −A + 2A′ + A2

)
ϕ

h
.

Hence

(4.12) f =
A

h
w′′ + C1w

′ + C0w + ψ,

where

C1 = −
1 + 2A′

h
, C0 =

1 −A + 2A′ + A2

h
.

Substituting (4.12) into (4.1) , we obtain

A

h
w(5) +

4∑
i=0

φiw
(i) = −

(
ψ(3) + A (z) ψ

)
,

where φi (i = 0, · · · , 4) are meromorphic functions in ∆ with finite iterated p−order.
Suppose that all meromorphic solutions f 6≡ 0 of (4.1) are of infinite iterated
p−order and ρp+1 (f) = ρ. If ψ 6≡ 0, then by Lemma 2.1 we obtain

(4.13) λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) = +∞

and

(4.14) λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) = ρ.

Suppose that f is a meromorphic solution of (4.1) of finite iterated p−order such
that

ρp (f) > max{ρp (A) ,ρp (ϕ)} .
If ψ(3) + A (z) ψ 6≡ 0, then by Lemma 2.1 we obtain

λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) .

Finally, we can state the following two results without the additional conditions
h 6≡ 0 and ψ is not a solution of (4.1).

Theorem 4.1 Suppose that A (z) is analytic function in ∆ of finite iterated p−order
0 < ρp (A) < ∞ and 0 < τp (A) < ∞, and that dj (z) (j = 0, 1, 2, 3) are finite it-
erated p−order analytic functions in ∆ that are not all vanishing identically such
that

max{ρp (dj) (j = 0, 1, 2, 3)} < ρp (A) .
If f is a nontrivial solution of (4.1), then the differential polynomial

(4.15) gf = d3f
(3) + d2f

′′ + d1f
′ + d0f

satisfies

ρp (gf ) = ρp (f) = ∞
and

ρp (A) ≤ ρp+1 (gf ) = ρp+1 (f) ≤ ρM,p (A) .
Theorem 4.2 Under the assumptions of Theorem 4.1, let ϕ (z) 6≡ 0 be an analytic
function in ∆ with finite iterated p−order. If f is a nontrivial solution of (4.1) ,
then the differential polynomial gf = d3f

(3) + d2f
′′ + d1f

′ + d0f (d3 6≡ 0) satisfies

(4.16) λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) = ∞



PROPERTIES OF SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS 171

and

(4.17) ρp (A) ≤ λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) ≤ ρM,p (A) .
Remark 4.1 The results obtained in Theorems 4.1 and 4.2 and refinement of
Corollaries 1.1 and 1.2 respectively.

Proof of Theorem 4.1 Suppose that f is a nontrivial solution of (4.1). Then by
Lemma 2.2, we have

ρp (f) = ∞, ρp (A) ≤ ρp+1 (f) ≤ ρM,p (A) .
First, we suppose that d3 6≡ 0. By the same reasoning as before we obtain that

h =

∣∣∣∣∣∣
H0 H1 H2
H3 H4 H5
H6 H7 H8

∣∣∣∣∣∣ ,
where H0 = d0 − d3A, H1 = d1, H2 = d2,H3 = d′0 − (d2 + d′3) A − d3A′, H4 =
d0 + d

′
1 −d3A, H5 = d1 + d′2,H6 = d′′0 − (d1 + 2d′2 + d′′3 ) A− (d2 + 2d′3) A′ −d3A′′,

H7 = 2d
′
0 + d

′′
1 − (d2 + 2d′3) A− 2d3A′, H8 = d0 + 2d′1 + d′′2 −d3A. Then

h = (3d0d1d2 + 3d0d1d
′
3 + 3d0d2d

′
2 − 6d0d3d

′
1 + 3d1d2d

′
1 + 3d1d3d

′
0

+d0d2d
′′
3 − 2d0d3d

′′
2 + d1d2d

′′
2 + d1d3d

′′
1 + d2d3d

′′
0 + 2d0d

′
2d
′
3 + 2d1d

′
1d
′
3 − 4d2d

′
0d
′
3

+2d2d
′
1d
′
2 + 2d3d

′
0d
′
2 −d1d

′
2d
′′
3 + d1d

′
3d
′′
2 + d2d

′
1d
′′
3 −d2d

′′
1d
′
3 −d3d

′
1d
′′
2

+d3d
′
2d
′′
1 −d

3
1 − 3d

2
0d3 − 2d1(d

′
2)

2 − 3d21d
′
2 −2d3(d

′
1)

2 −d22d
′′
1 −d

2
1d
′′
3 − 3d

2
2d
′
0

)
A

+ (2d0d2d
′
3 + 2d0d3d

′
2 −d1d2d

′
2 + 2d1d3d

′
1 − 4d2d3d

′
0 + d1d3d

′′
2

−d2d3d′′1 − 2d1d
′
2d
′
3 + 2d2d

′
1d
′
3 + 3d0d1d3 + d0d

2
2 −d

2
1d2 + d

2
2d
′
1 − 2d

2
1d
′
3

)
A′

+(d2d3d
′
1 + d0d2d3 −d1d3d

′
2 −d

2
1d3)A

′′ + (2d2d3d
′
3 − 3d1d

2
3 + 2d

2
2d3 − 2d

2
3d
′
2)AA

′

+
(
d32 − 3d1d2d3 − 3d1d3d

′
3 − 3d2d3d

′
2 −d2d3d

′′
3 − 2d3d

′
2d
′
3

+3d0d
2
3 + 3d

2
3d
′
1 + 2d2(d

′
3)

2 + 3d22d
′
3 + d

2
3d
′′
2

)
A2

−d33A
3 + 2d2d

2
3(A
′)2 −d2d23AA

′′ − 3d0d1d′0 −d0d1d
′′
1 −d0d2d

′′
0 − 2d0d

′
0d
′
2

+d1d
′′
0d
′
2 + d2d

′
0d
′′
1 −d2d

′
1d
′′
0 + d

3
0 + 2d0(d

′
1)

2 + 3d20d
′
1 + 2d2(d

′
0)

2

+d21d
′′
0 + d

2
0d
′′
2 − 2d1d

′
0d
′
1 + d0d

′
1d
′′
2 −d0d

′
2d
′′
1 −d1d

′
0d
′′
2.

By d3 6≡ 0, A 6≡ 0 and Lemma 2.3, we have ρp (h) = ρp (A), hence h 6≡ 0. For the
cases (i) d3 ≡ 0, d2 6≡ 0; (ii) d3 ≡ 0,d2 ≡ 0 and d1 6≡ 0 by using a similar reasoning
as above we get h 6≡ 0. Finally, if d3 ≡ 0, d2 ≡ 0, d1 ≡ 0 and d0 6≡ 0, then we have
h = d30 6≡ 0. Hence h 6≡ 0. By h 6≡ 0, we obtain

f =
1

h

∣∣∣∣∣∣
gf d1 d2
g′f d0 + d

′
1 −d3A d1 + d′2

g′′f 2d
′
0 + d

′′
1 − (d2 + 2d′3) A− 2d3A′ d0 + 2d′1 + d′′2 −d3A

∣∣∣∣∣∣ ,
which we can write

(4.18) f =
1

h

(
D0gf + D1g

′
f + D2g

′′
f

)
,

where

D0 = (d1d2 − 2d0d3 + 2d1d′3 + d2d
′
2 − 3d3d

′
1 −d3d

′′
2 + 2d

′
2d
′
3) A

+ (2d1d3 + 2d3d
′
2) A

′ + A2d23 + 3d0d
′
1 − 2d1d

′
0 + d0d

′′
2 −d1d

′′
1



172 LATREUCH AND BELAÏDI

−2d′0d
′
2 + d

′
1d
′′
2 −d

′
2d
′′
1 + d

2
0 + 2(d

′
1)

2,

D1 =
(
d1d3 − 2d2d′3 −d

2
2

)
A + d2d

′′
1 −d0d1 − 2d1d

′
1 + 2d2d

′
0 −d1d

′′
2,

D2 = d2d3A + d
2
1 −d2d

′
1 + d1d

′
2 −d0d2.

If ρp (gf ) < +∞, then by (4.18) we obtain ρp (f) < +∞, and this is a contradiction.
Hence ρp (gf ) = ρp (f) = +∞.

Now, we prove that ρp+1 (gf ) = ρp+1 (f) . By (4.15), we get ρp+1 (gf ) ≤
ρp+1 (f) and by (4.18) we have ρp+1 (f) ≤ ρp+1 (gf ). This yield ρp (A) ≤ ρp+1 (gf ) =
ρp+1 (f) ≤ ρM,p (A) .

Proof of Theorem 4.2 By setting w = gf −ϕ in (4.18) , we have

(4.19) f =
1

h
(D0w + D1w

′ + D2w
′′) + ψ,

where

(4.20) ψ =
D2ϕ

′′ + D1ϕ
′ + D0ϕ

h
.

Since d3 6≡ 0, then h 6≡ 0. It follows by Theorem 4.1 that gf is of infinite iterated
p−order analytic function and ρp (A) ≤ ρp+1 (gf ) ≤ ρM,p (A) . Since ρp (ϕ) < ∞,
then we have ρp (w) = ρp (gf ) = ρp (f) = ∞ and ρp (A) ≤ ρp+1 (w) = ρp+1 (gf ) =
ρp+1 (f) ≤ ρM,p (A) . Substituting (4.19) into (4.1) , we obtain

D2
h
w(5) +

4∑
i=0

φiw
(i) = −

(
ψ(3) + A (z) ψ

)
,

where φi (i = 0, · · · , 4) are meromorphic functions in ∆ with finite iterated p−order.
We prove first that ψ 6≡ 0. Suppose that ψ ≡ 0, then (4.20) can be rewritten as
(4.21) D2ϕ

′′ + D1ϕ
′ + D0ϕ = 0

and by Lemma 2.3, we have

(4.22) ρ (D0) > max{ρ (D1) ,ρ (D2)} .
By (4.21) we obtain

D0 = −
(
D2

ϕ′′

ϕ
+ D1

ϕ′

ϕ

)
.

Since ρp (ϕ) = β < ∞, then by Lemma 2.4 we have

T (r,D0) ≤ T (r,D1) + T (r,D2) + O

(
expp−2

(
log

1

1 −r

)β+ε)
, r /∈ E,

where E ⊂ [0, 1) is a set with
∫
E

dr
1−r < ∞. Then, by using Lemma 2.6, we get

ρp (D0) ≤ max{ρp (D1) ,ρp (D2)} ,
which is a contradiction. It is clear now that ψ 6≡ 0 cannot be a solution of (4.1)
because ρp (ψ) < ∞. Then, by Lemma 2.1 we obtain

λp (w) = λp (w) = λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) = ∞
and

ρp (A) ≤ λp+1 (w) = λp+1 (w) = λp+1 (gf −ϕ)
= λp+1 (gf −ϕ) = ρp+1 (f) ≤ ρM,p (A) .



PROPERTIES OF SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS 173

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Department of Mathematics, Laboratory of Pure and Applied Mathematics, Univer-
sity of Mostaganem (UMAB), B. P. 227 Mostaganem-(Algeria)

∗Corresponding author