ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 1(2011), Pages 14- 26.

GROWTH OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS IN THE UNIT DISC

(COMMUNICATED BY FIORALBA CAKONI)

BENHARRAT BELA¨IDI

Abstract. In this paper, we investigate the growth of solutions of higher
order linear differential equations with analytic coefficients in the unit disc
∆ ={z∈_{C}:|z|<1}. We obtain four results which are similar to those in
the complex plane.

1. _{Introduction and statement of results}

The growth of solutions of the complex linear differential equation

f(n)+an−1(z)f(n−1)+...+a1(z)f

0

+a0(z)f = 0 (1.1)

is well understood when the coefficients aj are polynomials [16, 17, 22, 27] or of finite (iterated) order of growth in the complex plane [7, 14, 15, 21]. In particular, Wittich [27] showed that all solutions of (1.1) are entire functions of finite order if and only if all coefficients are polynomials, and Gundersen, Steinbart and Wang [17] listed all possible orders of solutions of (1.1) in terms of the degrees of the polynomial coefficients.

Recently, there has been an increasing interest in studying the growth of ana-lytic solutions of linear differential equations in the unit disc ∆ by making use of Nevanlinna theory. The analysis of slowly growing solutions have been studied in [13, 19, 25]. Fast growth of solutions are considered by [8, 9, 11, 12, 19, 20].

In this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna’s theory on the complex plane and in the unit disc ∆ ={z∈C:|z|<1}(see [18, 19, 24, 26, 28, 29]).

We need to give some definitions and discussions. Firstly, let us give definition
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 ∆ (i.e., see [12, 13]).

1991Mathematics Subject Classification. 34M10, 30D35.

Key words and phrases. Linear differential equations; analytic solutions; unit disc.

c

2011 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted April 25, 2010. Published December 13, 2010.

Definition 1.1. For a meromorphic function f in∆ let

D(f) := lim r→1−sup

T(r, f)

−log (1−r),

where T(r, f) is the characteristic function of Nevanlinna of f. If D(f)<∞, we say that f is of finite degree D(f) (or is non-admissible); if D(f) = ∞, we say thatf is of infinite degree (or is admissible). Iff is an analytic function in∆, and

DM(f) := lim r→1−sup

log+M(r, f)

−log (1−r) ,

in which M(r, f) = max

|z|=r|f(z)| is the maximum modulus function, then we say

that f is a function of finite degree DM(f) if DM(f) < ∞; otherwise, f is of

infinite degree.

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 [7, 21, 22]). Let us define
inductively, forr∈[0,1), exp_{1}r:=er_{and exp}

p+1r:= exp exppr

, p∈_{N}. We also
define for allrsufficiently large in (0,1),log_{1}r:= logrand log_{p}_{+1}r:= log log_{p}r

,
p ∈ _{N}. Moreover, we denote by exp_{0}r := r, log_{0}r := r, log_{−}_{1}r := exp_{1}r and
exp_{−}_{1}r:= log_{1}r.

Definition 1.2. [8, 11, 20] Let f be a meromorphic function in ∆. Then the iterated p−order off is defined by

ρp(f) = lim r→1−sup

log+_{p}T(r, f)

−log (1−r) (p>1 is an integer),

where log+_{1} x = log+x = max{logx,0}, log+_{p}_{+1}x = log+log+_{p} x. For p = 1, this
notation is called order and for p= 2hyper order [19, 23]. If f is analytic in∆,
then the iterated p−order off is defined by

ρM,p(f) = lim r→1−sup

log+_{p}_{+1}M(r, f)

−log (1−r) (p>1 is an integer).

Remark 1.3. It follows by M. Tsuji ([26], p. 205) that iff is an analytic function in∆, then we have the inequalities

ρ1(f)6ρM,1(f)6ρ1(f) + 1,

which are the best possible in the sense that there are analytic functions g and h such that ρM,1(g) =ρ1(g)andρM,1(h) =ρ1(h) + 1, see [13]. However, it follows

by Proposition 2.2.2 in [22]that ρM,p(f) =ρp(f)forp>2.

Definition 1.4. (see[11]) The growth index of the iterated order of a meromorphic function f(z)in ∆is defined by

i(f) =

0, iff is non-admissible,

min{j∈_{N}:ρj(f)<+∞} if f is admissible,
+∞, ifρj(f) = +∞for allj∈N.

iM(f) =

0, iff is non-admissible,

min{j∈N:ρM,j(f)<+∞} if f is admissible, +∞, if ρM,j(f) = +∞for allj ∈N.

Remark 1.5. If ρp(f) <∞ or i(f) 6p, then we say that f is of finite iterated

p−order; if ρp(f) = ∞ or i(f) > p, then we say that f is of infinite iterated

p−order. In particular, we say that f is of finite order if ρ1(f)<∞or i(f)61; f is of infinite order ifρ1(f) =∞ ori(f)>1.

In [2], the author and Hamouda have proved the following result.

Theorem 1.6. [2] Let B0(z), ..., Bn−1(z) with B0(z) 6≡ 0 be entire functions.

Suppose that there exist a sequence of complex numbers (zj)_{j}_{∈}_{N} with lim

j→+∞zj=∞

and three real numbersα, β andµ satisfying0_{6}β < α andµ >0 such that

|B0(zj)|>exp{α|zj| µ

}

and

|Bk(zj)|6exp{β|zj| µ

} (k= 1,2, ..., n−1)

asj→+∞. Then every solutionf 6≡0 of the equation

f(n)+Bn−1(z)f(n−1)+...+B1(z)f0+B0(z)f = 0

has an infinite order.

It is natural to ask what can be said about similar situations in the unit disc
∆. Forn_{>}2,we consider a linear differential equation of the form

f(n)+An−1(z)f(n−1)+...+A1(z)f

0

+A0(z)f = 0 (1.2)

whereA0(z), ..., An−1(z), A0(z)6≡0 are analytic functions in the unit disc ∆ =
{z ∈ _{C} : |z| < 1}. It is well-known that all solutions of the equation (1.2) are
analytic functions in ∆ and that there are exactlynlinearly independent solutions
of (1.2), (see [19]).

A question arises : What conditions onA0(z), ..., An−1(z) will guarantee that

every solution f 6≡0 of (1.2) has infinite order? Many authors have investigated the growth of the solutions of complex linear differential equations in C, see [2, 3,

4, 5, 6, 10, 15, 21, 22, 28, 29]. For the unit disc, there already exist many results [8, 9, 11, 12, 19, 20, 23], but the study is more difficult than that in the complex plane, because the efficient tool of Wiman-Valiron theory doesn’t work in the unit disc.

In the present paper, we investigate the growth of solutions of the equation (1.2). We also study the growth of solutions of non-homogeneous linear differential equa-tions. We give answer to the question posed above and we obtain four theorems which are analogous to those obtained by the author in the complex plane (see [2, 5]).

Theorem 1.7. LetA0(z)6≡0, ..., An−1(z)be analytic functions in the unit disc∆.

Suppose that there exist a sequence of complex numbers(zj)_{j}_{∈}_{N}with lim

and three real numbersα, β andµ satisfying0_{6}β < α andµ >0 such that

|A0(zj)|>expp

α

_{1}

1− |zj|

µ

(1.3)

and

|Ak(zj)|6expp

β

_{1}

1− |zj|

µ

(k= 1, ..., n−1) (1.4)

asj→+∞,wherep_{>}1is an integer. Then every solution f 6≡0 of the equation
(1.2) satisfies ρp(f) =ρM,p(f) = +∞andρp+1(f) =ρM,p+1(f)>µ.

Settingp= 1 in Theorem 1.7, we deduce the following result.

Corollary 1.8. Let A0(z)6≡0, ..., An−1(z) be analytic functions in the unit disc

∆.Suppose that there exist a sequence of complex numbers(zj)_{j}_{∈}_{N}with lim
j→+∞|zj|=
1− and three real numbersα, β andµsatisfying 0_{6}β < αandµ >0 such that

|A0(zj)|>exp

α

_{1}

1− |zj|

µ

and

|Ak(zj)|6exp

β

1 1− |zj|

µ

(k= 1, ..., n−1)

as j →+∞. Then every solution f 6≡ 0 of the equation (1.2) satisfies ρ1(f) = ρM,1(f) = +∞ andρ2(f) =ρM,2(f)>µ.

Theorem 1.9. LetA0(z)6≡0, ..., An−1(z)be analytic functions in the unit disc∆

withi(A0) =p(16p <+∞)andmax{ρM,p(Ak) :k= 1, ..., n−1}6ρM,p(A0) =
ρ.Suppose that there exist a sequence of complex numbers (zj)_{j}_{∈}_{N}with lim

j→+∞|zj|=
1− and three real numbersα, βandµsatisfying0_{6}β < αandµ >0 such that for
any given ε >0 sufficiently small, we have

|A0(zj)|>expp

(

α

_{1}

1− |zj|

ρ−ε)

(1.5)

and

|Ak(zj)|6expp

(

β

_{1}

1− |zj|

ρ−ε)

(k= 1, ..., n−1) (1.6)

as j →+∞. Then every solution f 6≡ 0 of the equation (1.2) satisfies ρp(f) =

ρM,p(f) = +∞andρp+1(f) =ρM,p+1(f) =ρM,p(A0) =ρ.

From Theorem 1.9, by settingp= 1 we obtain the following result.

Corollary 1.10. Let A0(z)6≡0, ..., An−1(z)be analytic functions in the unit disc

∆ with max{ρM,1(Ak) : k = 1, ..., n−1} 6 ρM,1(A0) = ρ < +∞. Suppose that

there exist a sequence of complex numbers (zj)j∈N with _{j}_{→}lim_{+}_{∞}|zj|= 1− and three

real numbers α, β andµ satisfying 0 _{6}β < α and µ >0 such that for any given
ε >0 sufficiently small, we have

|A0(zj)|>exp

(

α

1 1− |zj|

and

|Ak(zj)|6exp

(

β

_{1}

1− |zj|

ρ−ε)

(k= 1, ..., n−1)

as j →+∞. Then every solution f 6≡ 0 of the equation (1.2) satisfies ρ1(f) = ρM,1(f) = +∞ andρ2(f) =ρM,2(f) =ρM,1(A0) =ρ.

In the next, we study the growth of non-homogeneous linear differential equation

f(n)+An−1(z)f(n−1)+...+A1(z)f

0

+A0(z)f =F(z) (n>2), (1.7)

whereA0(z), ..., An−1(z), F(z) are analytic functions in the unit disc ∆ ={z∈

C:|z|<1}. We obtain the following results:

Theorem 1.11. Let A0(z), ..., An−1(z) and F(z) be analytic functions in the

unit disc ∆ such that for some integer s, 1 _{6} s _{6} n−1, satisfying i(As) = p
(1_{6}p <∞) andmax{ρp(Aj) (j=6 s), ρp(F)}< ρp(As). Then every admissible

solution f of (1.7) withρp(f)<+∞ satisfiesρp(f)>ρp(As).

Settingp= 1 in Theorem 1.11, we deduce the following result.

Corollary 1.12. Let A0(z), ..., An−1(z) and F(z) be analytic functions in the

unit disc ∆ such that for some integer s, 1 _{6} s _{6} k−1 satisfying max{ρ1(Aj)
(j6=s), ρ1(F)} < ρ1(As)<+∞. Then every admissible solutionf of (1.7) with

ρ1(f)<+∞satisfiesρ1(f)>ρ1(As).

Theorem 1.13. LetA0(z), ..., An−1(z)andF(z)be analytic functions in the unit

disc ∆ such that for some integer s, 0 _{6}s _{6}k−1, we have ρp(As) = +∞ and
max{ρp(Aj) (j6=s), ρp(F)}<+∞ (16p <∞). Then every solution f of (1.7)

satisfiesρp(f) = +∞.

From Theorem 1.13, by settingp= 1 we obtain the following result.

Corollary 1.14. Let A0(z), ..., An−1(z) and F(z) be analytic functions in the

unit disc ∆ such that for some integer s, 0 _{6}s _{6}k−1, we have ρ1(As) = +∞

andmax{ρ1(Aj) (j 6=s), ρ1(F)}<+∞. Then every solution f of (1.7) satisfies ρ1(f) = +∞.

Remark 1.15. A similar results to Theorem 1.11 and Theorem 1.13 in the plane case were obtained by the author in[5].

2. _{Some Lemmas}

In this section we give some lemmas which are used in the proofs of our theo-rems.

Lemma 2.1. ([13], Theorem 3.1) Letkandj be integers satisfyingk > j _{>}0, and
letε >0andd∈(0,1). Iff is a meromorphic in∆ such thatf(j) does not vanish
identically, then

f(k)_{(}_{z}_{)}
f(j)_{(}_{z}_{)}

6

_{1}

1− |z|

2+ε

max

log 1

1− |z|, T(s(|z|), f)

!k−j

(|z|∈/E1),

where E1 ⊂ [0,1) is a set with finite logarithmic measure

R

E1 dr

1−r < +∞ and

s(|z|) = 1−d(1− |z|).Moreover, ifρ1(f)<+∞, then

f(k)_{(}_{z}_{)}
f(j)_{(}_{z}_{)}

6

_{1}

1− |z|

(k−j)(ρ1(f)+2+ε)

(|z|∈/ E1), (2.2)

while if ρp(f)<+∞for somep>2, then

f(k)_{(}_{z}_{)}
f(j)_{(}_{z}_{)}

6

exp_{p}_{−}_{1}

(_{}

1 1− |z|

ρp(f)+ε)

(|z|∈/E1). (2.3)

Lemma 2.2. ([19]) Letf be a meromorphic function in the unit disc ∆, and let
k_{>}1 be an integer. Then

m

r,f (k)

f

=S(r, f), (2.4)

whereS(r, f) =Olog+T(r, f) + log(_{1}_{−}1_{r}), possibly outside a setE2⊂[0,1)with

R

E2 dr

1−r <+∞. If f is of finite order (namely, finite iterated 1−order) of growth,

then

m

r,f (k)

f

=O

log( 1 1−r)

. (2.5)

In the next, we give the generalized logarithmic derivative lemma.

Lemma 2.3. Letf be a meromorphic function in the unit disc∆ for whichi(f) =

p_{>}1 andρp(f) =β <+∞, and letk>1 be an integer. Then for anyε >0,

m

r,f (k)

f

=O exp_{p}_{−}_{2}

_{1}

1−r

β+ε!

(2.6)

holds for allr outside a setE3⊂[0,1) with

R

E3 dr

1−r <+∞.

Proof. First fork= 1.Sinceρp(f) =β <+∞,we have for allr→1−

T(r, f)_{6}exp_{p}_{−}_{1}

_{1}

1−r

β+ε

. (2.7)

By Lemma 2.2, we have

m r,f f

0!

=O

ln+T(r, f) + ln( 1 1−r)

(2.8)

holds for allroutside a setE3⊂[0,1) with

R

E3 dr

1−r <+∞.Hence, we have

m r,f f

0!

=O exp_{p}_{−}_{2}

_{1}

1−r

β+ε!

, r /∈E3. (2.9)

Next, we assume that we have

m

r,f (k)

f

=O exp_{p}_{−}_{2}

1 1−r

β+ε!

, r /∈E3 (2.10)

for some an integerk_{>}1.SinceN r, f(k)

Tr, f(k)=mr, f(k)+Nr, f(k)

6m

r,f (k)

f

+m(r, f) + (k+ 1)N(r, f)

6m

r,f (k)

f

+ (k+ 1)T(r, f)

=O exp_{p}_{−}_{2}

_{1}

1−r

β+ε!

+ (k+ 1)T(r, f) =O exp_{p}_{−}_{1}

_{1}

1−r

β+ε!

.

(2.11) By (2.8) and (2.11), we again obtain

m

r,f (k+1)

f(k)

=O exp_{p}_{−}_{2}

_{1}

1−r

β+ε!

, r /∈E3 (2.12)

and hence,

m

r,f (k+1)

f

6m

r,f (k+1)

f(k)

+m

r,f (k)

f

=O expp−2

_{1}

1−r

β+ε!

, r /∈E3. (2.13)

Lemma 2.4. ([1]) Let g : (0,1)→R andh: (0,1)→R be monotone increasing
functions such that g(r) _{6} h(r) holds outside of an exceptional set E4 ⊂ [0,1)

of finite logarithmic measure. Then there exists a d ∈ (0,1) such that if s(r) =
1−d(1−r), theng(r)_{6}h(s(r))for allr∈[0,1).

Lemma 2.5. ([20]) LetA0(z), ..., An−1(z) be analytic functions in the unit disc

∆. Then every solutionf 6≡0 of (1.2) satisfies

ρM,p+1(f)6max{ρM,p(Ak) :k= 0,1, ..., n−1}. (2.14)

Proof of Theorem 1.7. Suppose thatf 6≡0 is a solution of (1.2) withρp(f)<+∞. Then by Lemma 2.1, there exists a setE1⊂[0,1) with finite logarithmic measure

R

E1 dr

1−r <+∞such that for allzsatisfying|z|∈/E1and fork= 1,2, ..., n, we have

f(k)(z)

f(z)

6 1 1− |z|

k(ρ1(f)+2+ε)

(|z|∈/E1), (2.15)

ifρ1(f)<+∞,while ifρp(f)<+∞for somep>2, then

f(k)(z)

f(z)

6

exp_{p}_{−}_{1}

(

_{1}

1− |z|

ρp(f)+ε)

(|z|∈/ E1). (2.16)

By (1.2), we can write

1_{6} 1

|A0(z)|

f(n) f +

An−1(z) A0(z)

f(n−1) f +...+

A1(z) A0(z)

Then from (1.3), (1.4), (2.15), (2.16) and (2.17), we next conclude that

1_{6} 1

expnα 1 1−|zj|

µo

_{1}

1− |zj|

n(ρ1(f)+2+ε)

+ exp

(β−α)

_{1}

1− |zj|

µ _{1}

1− |zj|

(n−1)(ρ1(f)+2+ε)

+...+ exp

(β−α)

_{1}

1− |zj|

µ _{1}

1− |zj|

ρ1(f)+2+ε

6n

_{1}

1− |zj|

n(ρ1(f)+2+ε)

exp

(β−α)

_{1}

1− |zj|

µ

(2.18)

or

1_{6} 1

expp

n

α_{1}_{−|}1_{z}
j|

µoexpp−1
(_{}

1 1− |zj|

ρp(f)+ε)

+

exp_{p}nβ_{1}_{−|}1_{z}
j|

µo

exp_{p}nα_{1}_{−|}1_{z}
j|

µoexpp−1
(_{}

1 1− |zj|

ρp(f)+ε)

+...+

exp_{p}nβ_{1}_{−|}1_{z}
j|

µo

exp_{p}nα_{1}_{−|}1_{z}
j|

µoexpp−1
(_{}

1 1− |zj|

ρp(f)+ε)

6n

exp_{p}nβ_{1}_{−|}1_{z}
j|

µo

exp_{p}nα_{1}_{−|}1_{z}
j|

µoexpp−1

(_{}

1 1− |zj|

ρp(f)+ε)

(2.19)

holds all zj satisfying|zj|∈/ E1 as |zj| →1−, both the limits of the right hand of inequalities (2.18) and (2.19) are zero as |zj| →1−. Thus, we get a contradiction. Henceρp(f) =ρM,p(f) = +∞.

Now letf 6≡0 be a solution of (1.2) withρp(f) = +∞.By Lemma 2.1, there exist s(|z|) = 1−d(1− |z|) and a set E2 ⊂[0,1) with finite logarithmic measure

R

E2 dr

1−r <+∞such that forr=|z|∈/E2, we have

f(k)(z)

f(z)

6 1 1− |z|

2+ε

max

log 1

1− |z|, T(s(|z|), f)

!k

(k= 1, ..., n).

(2.20) By (1.2), we can write

|A0(z)|6

f(n) f

+|An−1(z)|

f(n−1) f

+...+|A0(z)|

f0 f . (2.21)

Again from the conditions of Theorem 1.7, for all zj satisfying |zj| =rj ∈/ E2 as |zj| →1−, we get from (1.3), (1.4), (2.20) and (2.21) that

exp_{p}

α

_{1}

1− |zj|

µ

6|A0(zj)|6nexpp

β

_{1}

1− |zj|

×

_{1}

1− |zj|

2+ε

max

log 1 1− |zj|

, T(s(|zj|), f)

!n

. (2.22)

Noting thatα > β_{>}0, it follows from (2.22) that

exp

α(1−γ) expp−1

1 1− |zj|

µ

6n

_{1}

1− |zj|

n(2+ε)

Tn(s(|zj|), f) (2.23)

holds for allzj satisfying|zj|=rj∈/E2 as|zj| →1−,whereγ(0< γ <1) is a real number. Hence by Lemma 2.4 and (2.23), we obtain

ρp+1(f) =ρM,p+1(f) = lim rj→1−

suplog

+

p+1T(rj, f) −log (1−rj) >

µ.

Theorem 1.7 is thus proved. _{}

Proof of Theorem 1.9. Suppose thatf 6≡0 is a solution of equation (1.2). Using the same proof as in Theorem 1.7, we getρp(f) =ρM,p(f) = +∞.

Now we prove that ρp+1(f) = ρM,p(A0) = ρ. By Theorem 1.7, we have ρp+1(f)≥ρ−ε, and since ε >0 is arbitrary, we getρp+1(f)≥ρ. On the other

hand, from Lemma 2.5, we have

ρp+1(f) =ρM,p+1(f)6max{ρM,p(Ak) :k= 0,1, ..., n−1}

=ρM,p(A0) =ρ.

It yieldsρp+1(f) =ρM,p+1(f) =ρM,p(A0) =ρ.

Proof of Theorem 1.11. Let max{ρp(Aj) ( j6=s), ρp(F)} = β < ρp(As) = α. Suppose thatf is an admissible solution of (1.7) withρ=ρp(f)<+∞. It follows from (1.7) that

As(z) =

F f(s) −

_{f}(n)

f(s) +An−1(z) f(n−1)

f(s) +...+As+1(z) f(s+1)

f(s)

+As−1(z) f(s−1)

f(s) +...+A1(z) f0

f(s)+A0(z) f f(s)

= F

f(s) − f f(s)

_{f}(n)

f +An−1(z) f(n−1)

f +...+As+1(z) f(s+1)

f

+As−1(z) f(s−1)

f +...+A1(z) f0

f +A0(z)

. (2.24)

By Lemma 2.3 and (2.24), we obtain

T(r, As) =m(r, As)6m(r, F) +m

r, 1 f(s)

+m

r, f f(s)

+ n−1

X

j=0 j6=s

m(r, Aj) +O expp−2

_{1}

1−r

6T(r, F) +T

r, 1 f(s)

+T

r, f f(s)

+ n−1

X

j=0 j6=s

T(r, Aj)

+O expp−2

1 1−r

ρ+ε!

(2.25)

holds for allroutside a setE3⊂[0,1) with

R

E3 dr

1−r <+∞.Noting that

T

r, f f(s)

6T(r, f) +T

r, 1 f(s)

=T(r, f) +Tr, f(s)+O(1)

6(s+ 2)T(r, f) +o(T(r, f)) +O(1). (2.26) Thus, by (2.25), (2.26), we have

T(r, As)6T(r, F) +cT(r, f) + n−1

X

j=0 j6=s

T(r, Aj)

+O exp_{p}_{−}_{2}

1 1−r

ρ+ε!

(r /∈E3), (2.27)

where c >0 is a constant. Sinceρp(As) =α, then there exists

n

r0_{n}o r_{n}0 →1−
such that

lim
r0_{n}→1−

log+_{p} Tr0_{n}, As

−log (1−r0n)

=α. (2.28)

Set R_{E}

3 dr

1−r := logγ <+∞. Since

R1−

1−r0_{n}
γ+1
r0_{n}

dr

1−r = log (γ+ 1), then there exists a pointrn∈

r0_{n},1−1−r

0

n γ+1

−E3⊂[0,1). From

log+_{p} T(rn, As)

log_{1}_{−}1_{r}

n >

log+_{p} Tr0_{n}, As

logγ+1

1−r0_{n}

=

log+_{p} Tr0_{n}, As

log_{1}1

−r0_{n}+ log (γ+ 1)

, (2.29)

it follows that

lim rn→1−

log+_{p} T(rn, As)

log 1 1−rn

=α. (2.30)

So for any givenε(0<2ε < α−β),and forj 6=s

T(rn, As)>expp−1

_{1}

1−rn

α−ε

(2.31)

T(rn, Aj)6expp−1

_{1}

1−rn

β+ε

, T(rn, F)6expp−1

_{1}

1−rn

β+ε

(2.32)

hold forrn→1−.By (2.27), (2.31) and (2.32) we obtain forrn→1−

exp_{p}_{−}_{1}

_{1}

1−rn

α−ε

6nexp_{p}_{−}_{1}

_{1}

1−rn

β+ε

+O exp_{p}_{−}_{2}

_{1}

1−rn

ρ+ε!

. (2.33)

Therefore,

lim rn→1−

sup log

+

p T(rn, f) −log (1−rn) >

α−ε (2.34)

and since ε > 0 is arbitrary, we get ρp(f)> ρp(As) =α. This proves Theorem

1.11. _{}

Proof of Theorem 1.13. Setting max{ρp(Aj) (j =6 s), ρp(F)}=β,then for a given

ε >0, we have

T(r, Aj)6expp−1

_{1}

1−r

β+ε

(j6=s), T(r, F)_{6}exp_{p}_{−}_{1}

_{1}

1−r

β+ε

(2.35)

forr→1−.Now by (2.24), we have

T(r, As) =m(r, As)6m(r, F) +m

r, 1 f(s)

+m

r, f f(s)

+ n−1

X

j=0 j6=s

m(r, Aj) + n

X

j=1 j6=s

m

r,f (j)

f

6T(r, F) +T

r, 1 f(s)

+T

r, f f(s)

+ n−1

X

j=0 j6=s

T(r, Aj) + n

X

j=1 j6=s

m

r,f (j)

f

. (2.36)

Hence by (2.26) and (2.36) we obtain that

T(r, As)6T(r, F) +cT(r, f) + n−1

X

j=0 j6=s

T(r, Aj) + n

X

j=1 j6=s

m

r,f (j)

f

, (2.37)

wherec >0 is a constant. Ifρ=ρp(f)<+∞, then by Lemma 2.3

m

r,f (j)

f

=O exp_{p}_{−}_{2}

_{1}

1−r

ρ+ε!

(j= 1, ..., n;j 6=s) (2.38)

holds for allroutside a setE3⊂[0,1) with

R

E3 dr

1−r <+∞.Forr→1

−_{,}_{we have}

T(r, f)_{6}expp−1

_{1}

1−r

ρ+ε

. (2.39)

Thus, by (2.35), (2.37), (2.38) and (2.39), we get

T(r, As)6nexpp−1

_{1}

1−r

β+ε

+cexp_{p}_{−}_{1}

_{1}

1−r

ρ+ε

+O exp_{p}_{−}_{2}

1 1−r

ρ+ε!

(2.40)

forr /∈E3 andr→1−. By Lemma 2.4, we have for anyd∈(0,1)

T(r, As)6nexpp−1

_{1}

d(1−r)

β+ε

+cexp_{p}_{−}_{1}

_{1}

d(1−r)

+O exp_{p}_{−}_{2}

_{1}

d(1−r)

ρ+ε!

(2.41)

forr→1−.Therefore,

ρp(As)6max{β+ε, ρ+ε}<+∞. (2.42)

This contradicts the fact thatρp(As) = +∞. Hence ρp(f) = +∞.

Acknowledgments. The author would like to thank the referee for his/her helpful remarks and suggestions to improve this article.

References

[1] S. Bank,General theorem concerning the growth of solutions of first-order algebraic

differ-ential equations, Compositio Math.25(1972), 61–70.

[2] B. Bela¨ıdi, S. Hamouda, Growth of solutions of n-th order linear differential equation with

entire coefficients, Kodai Math. J.25(2002), 240-245.

[3] B. Bela¨ıdi, On the iterated order and the fixed points of entire solutions of some complex linear differential equations, Electron. J. Qual. Theory Differ. Equ. 2006, No. 9, 1-11. [4] B. Bela¨ıdi,On the meromorphic solutions of linear differential equations, J. Syst. Sci.

Com-plex.20(2007), no. 1, 41–46.

[5] B. Bela¨ıdi, Iterated order of fast growth solutions of linear differential equations, Aust. J. Math. Anal. Appl.4(2007), no. 1, Art. 20, 1-8.

[6] B. Bela¨ıdi,Growth and oscillation theory of solutions of some linear differential equations, Mat. Vesnik60(2008), no. 4, 233–246.

[7] L. G. Bernal,On growth k-order of solutions of a complex homogeneous linear differential

equation, Proc. Amer. Math. Soc.101(1987), no. 2, 317–322.

[8] T. B. Cao, H. X. Yi,The growth of solutions of linear differential equations with coefficients

of iterated order in the unit disc, J. Math. Anal. Appl.319(2006), 278-294.

[9] T. B. Cao, H. X. Yi,On the complex oscillation of second order linear differential equations

with analytic coefficients in the unit disc, Chinese Ann. Math. Ser. A28(5) (2007), 403–416

(in Chinese).

[10] T. B. Cao, H. X. Yi,On the complex oscillation of higher order linear differential equations

with meromorphic functions, J. Syst. Sci. Complex.20(2007), N◦_{1, 135-148.}

[11] T. B. Cao,The growth, oscillation and fixed points of solutions of complex linear differential

equations in the unit disc, J. Math. Anal. Appl.352(2009), 739–748.

[12] Z. X. Chen, K.H. Shon,The growth of solutions of differential equations with coefficients of

small growth in the disc, J. Math. Anal. Appl.297(2004), 285–304.

[13] I. E. Chyzhykov, G. G. Gundersen, J. Heittokangas,Linear differential equations and

loga-rithmic derivative estimates, Proc. London Math. Soc.86(2003), 735–754.

[14] M. Frei,Uber die L¨¨ osungen linearer Differentialgleichungen mit ganzen Funktionen als

Ko-effizienten, Comment. Math. Helv.35(1961), 201-222.

[15] G. G. Gundersen,Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc.305(1988), no. 1, 415–429.

[16] G.G. Gundersen, E.M. Steinbart,Finite order solutions of nonhomogeneous linear differential

equations, Ann. Acad. Sci. Fenn. Ser. A I Math.17(2) (1992), 327–341.

[17] G.G. Gundersen, E.M. Steinbart, S. Wang, The possible orders of solutions of linear

dif-ferential equations with polynomial coefficients, Trans. Amer. Math. Soc. 350(3) (1998),

1225–1247.

[18] W. K. Hayman,Meromorphic functions, Oxford Mathematical Monographs Clarendon Press, Oxford, 1964.

[19] J. Heittokangas,On complex linear differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss.122(2000), 1-54.

[20] J. Heittokangas, R. Korhonen, J. R¨atty¨a,Fast growing solutions of linear differential

equa-tions in the unit disc, Results Math.49(2006), 265-278.

[22] I. Laine,Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, New York, 1993.

[23] Y. Z. Li,On the growth of the solution of two-order differential equations in the unit disc, Pure Appl. Math.4(2002), 295–300.

[24] R. Nevanlinna,Eindeutige analytische Funktionen, Zweite Auflage. Reprint. Die Grundlehren der mathematischen Wissenschaften, Band 46. Springer-Verlag, Berlin-New York, 1974. [25] D. Shea, L. Sons,Value distribution theory for meromorphic functions of slow growth in the

disc, Houston J. Math.12(2) (1986), 249-266.

[26] M. Tsuji,Potential Theory in Modern Function Theory, Chelsea, New York, (1975), reprint of the 1959 edition.

[27] H. Wittich, Zur Theorie linearer Differentialgleichungen im Komplexen, Ann. Acad. Sci. Fenn. Ser. A I Math.379(1966), 1–18.

[28] L. Yang,Value Distribution Theory, Springer-Verlag, Berlin, 1993.

[29] H. X. Yi, C. C. Yang,Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557. Kluwer Academic Publishers Group, Dordrecht, 2003.

Benharrat Bela¨ıdi

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