On the growth of solutions of some higher order linear differential equations with meromorphic coefficients Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 10. № 1 (2018). С. 118-136.

ON THE GROWTH OF SOLUTIONS OF SOME HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS

M. SAIDANI, B. BELAÏDI

Abstract. In this paper, by using the value distribution theory, we study the growth and the oscillation of meromorphic solutions of the linear differential equation

f(k) + [Ak-1A(z)ep"-l(z) + A^z^-^ f(k-1)

+ ••• + [Ao,i(z)ePo(z) + Ao,2(z)eQo(z)) f = F(z),

where Aj^(z) 0) (j = 0,... ,k — 1), F(z) are meromorphic functions of a finite order, and Pj (z),Qj (z) (j = 0,1,...,k — 1; i = 1, 2) are polynomials with degree n ^ 1. Under some conditions, we prove that as F = 0 each meromorphic solution f ^ 0 with poles of uniformly bounded multiplicity is of infinite order and satisfies p2(f) = n and as F ^ 0 there ^^^^^s at most one exceptional solution f0 of a finite order, and all other transcendental meromorphic solutions f with poles of uniformly bounded multiplicities satisfy X(f) = X(f) = p (f) = rnd X2 (f) = A2 (f) = p2 (f) ^ max {n, p (F)} . Our results extend the previous results due Zhan and Xiao [19].

Keywords: Order of growth, hvper-order, exponent of convergence of zero sequence, differential equation, meromorphic function.

Mathematics Subject Classification: 34M10, 30D35

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, see [12], [18]. Let p (/) stands for the order of growth of a meromorphic function f and the hvper-order of f is defined by

m v log log T (rj)

P2 (/) = limsup---,

log r

where T (r, f) is ^te Nevanlinna characteristic function of f , see [12], [14], [18].

Definition 1.1. f|15], [17]J Let f be a meromorphic function. The convergence exponent of the zero-sequence of a meromorphic function f is defined by

log N (r, j)

A (f) = limsup--—--—,

log r

M. Saidani and B. Belaidi, On The Growth of Solutions of Some Higher Order Linear Differential Equations With Meromorphic Coefficients. © Saidani M., Belaidi B. 2018. Поступила 6 января 2017 г.

where N yr, j J is the integrated counting function of zeros of f in {z : |z | ^ r], and the exponent of convergence the sequence of distinct zeros of f is defined by

_ log N (r,

A (f) = lim sup--—--—,

log r

where N ^r, jj is the integrated counting function of distinct zeros of f in {z : |z| ^ r] . The

hyper convergence exponents of the zero-sequence and the distinct zeros of f are defined respectively by

log log N (r, J j _ log log N (r, J j

\2 (f ) = lim sup-----—, A2 (f ) = lim sup-----—.

log r log r

Several authors [3], [9], [14] have study the growth of solutions of the second order linear differential equation

f" + A1(z)ep (z)f' + A2{z)eQ(z)f = 0, (1.1)

where P(z), Q(z) are nonconstant polynomials, A1(z), A2(z) (^ 0) are entire functions such that p (A1) < degP(z), p (A2) < degQ(z). Gundersen showed in [9] that if degP(z) = degQ(z), then each nonconstant solution of (1,1) is of infinite order. If deg P(z) = deg Q(z), then (1,1) mav have nonconstant solutions of a finite order. For instance f (z) = e^ + 1 satisfies f'' + ez f' — ez f = 0.

In [10], Habib and Belaidi studied the order and hvper-order of solutions of some higher order linear differential equations and they proved the following result.

Theorem 1.1. (¡10}) Let Aj (z) (^ 0), (j = 1, 2), Bt (z)& 0) (I = 1,...,k - Dm (m = 0,... ,k — 1) be entire functions with

max {p (Aj) ,p (Bi) ,p (Dm)] < 1,

bi (I = 1,... ,k — 1) be complex constants such that (i) arg bi = arg a1 an d bi = Q a1 (0 < Q < 1) (/ E h) and (ii) bi is a real constant such that bi ^ 0 (/ E I2), where I1 = I2 = I1HI2 = 0, I1 U I2 = {1, 2,...,k — 1], an d a1} a2 are complex numbers su ch, that a1a2 = 0 a1 = a2 (suppose that |a1| ^ If arga1 = n or a1 is a real number such that a1 < ^, where

c = max {ci : I E I1] and b = min {bi : I E I2], then each solution f ^ 0 of the equation

f(k) + (Dk-1 + Bk-1ebk-lZ) f(k-1) + ... + (D1 + B1eblZ) f'

+ (Do + A1eaiz + ^e^) f = 0 ( ' j

satisfies p (f) = and p2 (f) = 1.

And in [2], they studied the order and hvper-order of solutions of some higher order linear differential equations with meromorphic coefficient and they proved the following result.

Theorem 1.2. (\2\) Let Aj (z) (^ 0) (j = 1, 2 ), Bt(z) (^ 0) (I = 1,...,k — 1) be meromorphic functions with

max {p (Aj) (j = 1, 2) ,p (Bl) (I = 1,...,k — 1)] < 1,

bi (I = 1,... ,k — 1 ) be complex constants such that (i) bi = cia1 (0 < Q < 1) (I E 11) and (ii) bi is a real constant sue h that bi < 0 (I E I2), whe re I1 = 0, I2 = 0, I1 H I2 = 0, I1 U I2 = {1, 2,...,k — 1], an d a1} a2 are complex numbers su ch, that a1a2 = 0 a1 = a2 (suppose that |a1| ^ I^D- If arga1 = n or a1 is a real number such that a1 < where c = max {ci,1 E h] and b = min {bi,l E I2], then each meromorphic solution f (^ 0) with poles of uniformly bounded multiplicities of the equation

f(k) + Bk-1ebk-lZ f(k-1) + ■ ■ ■ + BxehlZ f' + (A1eaiz + ^e"2") f = 0 (1.3)

satisfies p (f) = and p2 (f) = 1.

In [19], Zhan and Xiao studied the homogeneous and nonhomogeneous higher order differential equations and obtained the following results.

Theorem 1.3. ([19]) Let Aji(z) (^ 0) be entire functions with p (Aji) < n, n ^ 1 is a positive integer, j = 0,1,...,k — 1; i = 1,2. Let Pj (z) = a,j,nzn + ••• + aj,0 and Qj (z) = bj,nzn + ■ ■ ■ + bj,0 be polynomials, where a,j,q, bj,q (j = 0,1,... ,k — 1; q = 0,1,... ,n) are complex numbers such, that aj,nbj,n = 0, a0,n = b0,n and aj,n = Cja0,n,bj,n = Cjb0,n,Cj > 1, j = 1,... ,k — 1 are distinct numbers. Then each solution f (^ 0) of the equation

f(k) + (Ak-iA(z)ep*-l(z) + Ak-i,2(z)e^-l(z)) f(k-1) + ••• + (A0,i(z)ePo(z) + Ao,2(z)eQ°(z)) f = 0

of a finite order.

Theorem 1.4. (|19]J Let Aji(z) (^ 0) be entire functions with p (Aji) < n, where n ^ 1 is a positive integer, j = 0,1,...,k — 1; i = 1,2. Le t Pj (z) = a,j,nzn + ••• + aj,0 and Qj (z) = bj,nzn + ••. + bj,0 be polynomials, where a,j,q, bj,q (j = 0,1,... ,k — 1; q = 0,1,... ,n) are complex numbers such, that aj,nbj,n = 0, a0,n = b0,n and aj,n = Cja0,n,bj,n = Cjb0,n,Cj > 1, j = 1,... ,k — 1 are distinct nu,mbers. F(z0) is an entire function of a finite order. Then the equation

f(k) + (Ak-ij(z)ep*-1 (z) + Ak-it2(z)eQk-l(z)) f(k-1)

+ ••• + (A0,i(z)ePo(z) + Ao,2(z)eQo(z)) f = F(z) satisfies the following statements:

(i) There exists at most one exceptional solution f0 of a finite order, and all other solutions

satisfy X (f) = A (f) = p (f) = and % (f) = A2 (f) = p2 (J) ^ max {n., p (F)} .

(n) If there exists f0 of a finite order, then p (f0) ^ max{ n, X (f0) , p (F)} .

(Hi) If F(z) is an entire function of order less than n and arg a0,n = arg b0,n, then each, solution

of (1.5) is of infinite order.

In this paper, we are concerned with a more general problem. We extend and improve Theorem 1,3 and Theorem 1,4, In fact, we will prove the following theorems.

Theorem 1.5. Let Aji(z) (^ 0) be meromorphic functions of a finite order such that max{p (Aji) ,j = 0,1,...,k — 1; i = 1,2} < n, where n ^ 1 is a positive integer. Let Pj(z) = a,j,nzn + ••. + a,j,0 and Qj(z) = bj,nzn + ••• + bj,0 be polynomials, where a,j,q,bj,q (j = 0,1,... ,k — 1; q = 0,1,... ,n) are complex numbers su ch, that aj,nbj,n = 0, a0,n = b0,n and aj,n = cj a0,n, bj,n = Cj b0>n, Cj > 1, j = 1,... ,k — 1 are distinct numbers. Then each meromorphic solution f (^ 0) of equation (1-4) with poles of uniformly bounded multiplicity is of infinite order and satisfies p2(f) = n.

Theorem 1.6. Let Aji(z) (^ 0), F(z)(^ 0) be meromorphic functions of a finite order with max{p (Aji), j = 0,1,... ,k — 1; i = 1, 2} < n, where n ^ 1 is a positive integer. Let Pj(z) = a,j,nzn + ••. + a,j,0 and Qj(z) = bj,nzn + ••• + bj,0 be polynomials, where a,j,q,bj,q (j = 0,1,... ,k — 1; q = 0,1,... ,n) are complex numbers su ch, that aj,nbj,n = 0, a0,n = b0,n and aj,n = cja0,n, bj,n = Cjb0,n, Cj > 1, j = 1,... ,k — 1 are distinct numbers. Then the equation (1.5) satisfies:

(i) There exists at most one exceptional meromorphic solution f0 with finite order, and all other transcendental meromorphic solutions f with poles of uniformly bounded multiplicities satisfy

A (f ) = A (f ) = p (f ) = +»

and _

X2 (f) = A2 (f) = P2 (f) ^ max {n,p (F)} . (n) If there exists f0 of a finite order, then p (f0) ^ max{ n, X (f0) , p (F)} . (Hi) If F(z) is a meromorphic function of order less than n and arg a0,n = arg b0,n, then each, meromorphic solution f of (1.5) with poles of uniformly bounded multiplicities is of infinite order and satisfies p2(f) = n.

2. Auxiliary lemmata First, we recall the following definitions. The linear measure of a set E c [0, is defined

as

r

m (E) = / Xe (t)dt Jo

and the logarithmic measure of a set F c [1, is defined by

Im ( F) = i^dt,

Ji t

where xh (t) is the characteristic function of a set H.

Lemma 2.1. ([1]) Let Pj (z) (j = 0,1,..., k) be polynomials with deg P0 = n (n ^ 1) and deg Pj C n (j = 1,..., k) .Let Aj (z) (j = 0,1,... ,k) be meromorphie functions of a finite order and max {p ( Aj) , j = 0,1,... ,k} < n such that A0(z) ^ 0. We denote

F (z) = AkePk(z) + Ak-i ePk-l(z) + ■ ■ ■ + Ai ePl(z) + Age p°(z).

If deg ( P0(z) — Pj (z)) = n for alI j = 1,..., k, then F is a nontrivial meromophic function with finite order satisfying p (F) = n.

Lemma 2.2. (|8]J Let f (z) be a transcendental meromorphie function and let a > 1 and e > 0 be given constants. Then there exist a set E1 c (1, of a finite logarithmic measure and a constant B > 0 that depends only on a and positive integers (n,m) obeying n > m ^ 0 such, that for all z -satisfying \z\ = r ^ [0,1] U E1, we have

fin)(z)

f(m)(z)

C B

T(ar, f)

-(a-lf± (log" r) logT(ar, f)

Lemma 2.3. ([H]) Let P(z) = (a + if3) zn + ■ ■ ■ (a, ft are real numbers, \a\ + \f3\ = 0j be a polynomial with degree n ^ 1 and A(z) be a meromorphie function with p (A) < n. Let

f(z) = A(z)ep{z), z = reie, 5 (P, 9) = a cos n0 — ¡3 sin nd.

Then for any given e > 0, there exists a set E2 C [1, of a finite logarithmic measure such, that for each B e [0, 2tt) \H (H = {Be [0, 2tt) : 5 (P, 0) = 0}) and fo r \z\ = r / [0,1] U E2, r ^ we have

(i) if 5 (P, 9) > 0, then

exp {(1 — e)S ( P, 9) rn} C |f {reid) | C exp {(1 + e) 6 ( P, 9) rn} ,

(ii) if 8 (P, 9) < 0, then

exp {(1 + e) 6 ( P, 9) rn} C | f {reid) | C exp {(1 — e) S ( P, 9) rn} .

Lemma 2.4. (|5]J Let f(z) be a meromorphie function of order p (f) = p < Then

for any given e > 0, there exists a set E3 C (1, that has finite linear measure and finite logarithmic measure such, that as \z\ = r e [0,1] U E3, r ^ +x>, we have \ f (z)\ C exp (rp+£).

It is well known that due to the Wiman-Valiron theory [13], [15], it is important to studvt the properties of entire solutions of differential equations. In [4], Chen extended the Wiman-Valiron theory from entire functions to meromorphie functions. Here we give a special form of the result given by Wang and Yi in [17], when meromorphie function has infinite order,

<x

Let g(z) = an be an entire function. By p (r) = max{\an\ rn; n = 0,1,...} we denote

n=0

the maximum term of g and by ug (r) = max{m : p (r) = \am\ rm} we denote the central index

n—m

Lemma 2.5. (|17]J Let f (z) = g(z)/d(z) be a meromorphic function of infinite order obeying p2 (f) = o, g(z) and d(z) are entire functions, where p (d) < Then there exists a

}meN satisfying

sequence of complex numbers {z„

rm ^ dm G [0, 2t) ; m G N, lim dm = 90 G [0, 2n), \g (zm)\ = M (rm,g)

and for sufficiently large m we have

f (n)(zm) (vg (rmy

m—+<x

f (Zm) V ^m

lOglOg Ug (r m

lim sup

log rn

^ (1 + o (1)) (n G N), = p2(g) =

Lemma 2.6. ([9]) Let >p : [0, ^ R and ^ : [0, +rc>) ^ R be a monotone nondecreasing functions such that p(r) ^ ip(r) for a 11 r G (E4 U [0,1]), where E4 is a set of a finite logarithmic measure. Let a > 1 be a given constant. Then there exists an ri = ri(a) > 0 such that p(r) ^ ip(ar) for a 11 r > ri.

Lemma 2.7. Suppose that k ^ 2 rnd F, A0, Ai,... ,Ak-x are meromorphic functions such that p = max [p (Aj) j = 0,1, 2,... ,k — 1, p (F)} < +x>. Let f (z) be a transcendental meromorphic solution with all poles of f are of uniformly bounded multiplicity, of equation

f(k) + Ak-i f(k-i) + ... + Aif' + Ao f = F.

(2.1)

Then p2 (f) ^ p.

Proof. We assume that f is a transcendental meromorphic solution of equation (2.1). If p (f) < then p2 (f) = 0 ^ p. Assume that f is a meromorphic solution to equation (2.1) of infinite order with poles of uniformly bounded multiplicity. By (2.1) we have

f(k)

f

< \Ak-i(z)

f (k-i)

f

+ ■■■ + \Ai(z )\

+

+ \4>(*)\

(2.2)

By (2.1) it follows that the poles of f can locate only at the poles of Aj (j = 0,... ,k — 1) and F. Note that the poles of f are of uniformly bounded multiplicity. Hence, A (1/f) ^ p. By the Hadamard factorization theorem, we know that f can be expressed as f (z) = where g(z) and d(z) are entire functions with

A (d) = p (d) = A (1/f) ^ p<p (f) = p ($) = +«> and p2 (f) = P2 (g). By Lemma 2.5, there exists a sequence { Zm

}meN satisfyinS

rm ^ dm G [0, 2n), lim „m

lim 6r,

m—

such that for m sufficiently large we have

f (n)(Zm) ( Vg (Vm)'

f (Zm)

i "g (rm)\ \ Zm J

do G [0, 2^), \g(Zm)\ = M(rm,g)

(1 + o(1)) (n G N)

and

V loglog ^g (J'm) ( ,

lim sup---= p2(g).

log rm

(2.3)

(2.4)

By Lemma 2.4, for each given e > 0, there exists a set E3 C (1, of a finite logarithmic measure such that

\F(z)\ ^ exp {rp+£u , \d(z)\ ^ exp {rp+£} (2.5)

and

\Aj(z)\ ^ exp {rp+£) (j = 0,...,k — 1)

(2.6)

n

hold for Izl = r G [0,1] U E3, r ^ Since M (r, g) ^ 1 for r sufficiently large, it follows from (2.5) that

F (z)

lF(*)M^ _ lF(*)M^ < exp ) ,2 7)

lg(z)l _ M (r, g) < eXP|2f (2'7)

Substituting (2.3), (2.6) and (2.7) into (2.2), we obtain

(|1 + o(1)| < e^+i' |1 + 0(1)1 + er"+S + e2r• \ j . y v m j

j=i

It follows that

(ua(rm))k |1 + o(1)| < ( k + 1) e2ri+rkm (ua(rm))k~l |1 + o(1)| •

ug(rm) < ( k + 1) Arkme2r™ ', (2.8)

Hence,

„k ^2 riJ+

m

where the sequence {zm _ rme%dm}meN satisfies

rm G [0,1] UE3, rm ^ OmE [0, 2tt), lim dm _ 90 G [0, 2tt), |^(zm)| _M (r m, g)

and A > 0 is some constant. Then by (2.8), Lemma 2.6 and e > 0 being arbitrary, we obtain that p2(g) _ p2(f) < P- □

Remark 2.1. For F = 0, Lemma 2.7 was proved by Chen and Xu in [7].

( ) a( )

of g. For each, sufficiently large |z| _ r, let zr _ re%dr he a point satisfying ^ (zr)| _ M (r, g). Then there exist a constant 5r (> 0) and a set E5 of a finite logarithmic measure such, that for all z satisfying | z| _ r G E5 and argz _ 9 G [9r — 5r, 9r + 5r] , we have

g[n)(z) fug (r)

( )

(1 + o (1)) (n ^ 1 is an integer) •

Lemma 2.9. f|8]J Let f(z) be a transcendental meromorphic function of a finite order p. Let r = {( ki, ji) , ( k2, j 2) ,..., (km, jm)} denote a set of distinct pairs of integers satisfying ki > ji ^ 0 (i = 1,2,... ,m) and let £ > 0 be a given constant. Then there exists a set E6 c [1, +<x>) of a finite logarithmic measure such, that for all z obeying Izl = r G [0,1] U E6 and ( k, j) G r, we have

f(k)(z) ^ I^ik-j)(p-i+s)

f(3) ( z)

Lemma 2.10. Let f(z) = g(z)fd(z) be a meromorphic function with p(f) = p ^ ( ) ( )

()

(n) g, d are transcendental and X (d) = p (d) = 3 < p (g) = p.

For each sufficiently large Izl = r, let zr = reir be a point satisfying lg (zr)| = M (r, g) and let ug(r) be the central index of g. Then there exist a constant 5r (> 0), a sequence {rm}meN , i"m ^ and a set E7 of finite logarithmic measure such, that the estimation

f{n)(z) _( ^ (rm)

f(z)

^g (^m) J

(1 + o (1)) (n ^ 1 is an integer)

holds for all z satisfying Izl = rm G E7, rm ^ and arg z = 9 G [9r — 5r, 9r + 5r]. Proof. By mathematical induction, we obtain

. . n(n) ^aU) „ /d' \h f d(n) \jn

3=0 Uv3n) v 7 v 7

where Cjjr„jn are constants and j + ji + 2j2 + ■ ■ ■ + njn = n. Hence,

£ f (f)'...(^ r

J(n) g(n)

~T = 1T

(2.10)

j=0 (j V jn)

For each sufficiently large \z\ = r, let zr = rez&r be a point satisfying \g (zr)\ = M (r, g). By Lemma 2.8, there exist a constant 5r (> 0) and a set E5 of a finite logarithmic measure such that for all z obeying \ z\ = r G E5 and argz = 9 G [9r — 5r, 9r + 5r], we have

g(3)(z) _ f Vg (r) 9(z)

(1 + o(1)) (j = 1, 2,...,n)

(2.11)

where ug (r) is the central index of g. Substituting (2.11) into (2.10) yields

f(n)(z) ( va (r)

f(A

(1 + 0(1))

+ g (^ )'" » + •»» £ <",........(i )*■■■( ?)"'

(2.12)

We can choose a constant a such that P < a < p. By Lemma 2.9, for any given e (0 < 2e < a — P), we have

d(s) (z)

d(z)

^r s(l-i+e) ( s = 1, 2,...,n),

(2.13)

where \z\ = r G [0,1] U E6, E6 C (1, with lm (E6) < From this and

ji + 2 j2 + ■ ■ ■ + njn = n — j, we have

\ \

n-

(f )*■■■( £)j

^ w

(n-j)(l3+£)

(2.14)

for \ z\ = r G [0,1] U E6. By p (g) = p, there exists a sequence [r'm} (r'm ^ satisfying

(2.15)

lim log ug(rm) = p.

r'mlog r'm

Setting the logarithmic measure of E7 = [0,1] U E5 U E6, lm, (E7) = 5 < there exists a point rm G [^, (S + 1) r'm] \ E7. Since

log g( ) log g( m)

log g( m)

log rm log [(i + 1) r'm] (log r'm)

1+

log^+i)

log r'm

we get

log g( m)

lim

rm—+tx log T.

= P.

(2.16)

(2.17)

Hence, for sufficiently large m, we obtain

ug (rm) 2 2 C-, (2.18)

where p — e can be replaced by a large enough number M if p = This and (2.14) imply

j-n 'd!\n id(n)xjn

^ r'-^-^ ^ 0, ^ ^

(2.19)

where \z\ = rm G E7 and argz = 9 G [9r — 5r, 9r + 5r]. From (2.12) and (2.19), we obtain our result. □

Lemma 2.11. Let f(z) = g(z)/d(z) be a meromorphic function with p(f) = p ^ ( ) ( )

(i) g is transcendental and d is polynomial,

(n) g, d are transcendental and X (d) = p (d) = 3 < p (g) = p.

For each, sufficiently large I zl = r, let zr = reidr be a point satisfying lg (zr)| = M (r, g). Then there exist a constant 5r (> 0), a sequence {rm}meN, rm ^ and a set E8 of a finite

logarithmic measure such, that the estimate

f(z)

^ I""™ (n ^ 1 № an integer)

f(n)(z)

holds for all z satisfying I zI = rm G E8, rm ^ and argz = 9 G [9r — 5r, 9r + 5r].

Proof Let zr = reir be a point satisfying Ig (zr)I = M (r, g). By Lemma 2,10, there exist a constant 5r (> 0), a sequen ce {rm} me N , rm ^ and a set E8 of a finite logarithmic measure such that the estimate

fin)(Z) ( Ug (rm)

= ^ ZJ (1 + 0 (1)) (n ^ 1 is an integer) (2.20)

holds for all z satisfying IzI = rm G E8, rm ^ and argz = 9 G [9r — 5r, 9r + 5r]. On the other hand, for any given e > 0 and sufficiently large m we obtain

^g (rm) > r^, (2.21)

where p — e can be replaced by a large enough number M if p = Hence, we have

/(-)

mz) «^

This completes the proof. □

Lemma 2.12. ([12]) Let f be a meromorphic function and let k gN. Then

m(r, —^ = S (r, f),

where S (r, f) = 0 (logT (r, f) + log r), possibly outside a set E9 c (0, with a finite linear

( f k)\ m ir, ) = 0 (logr).

Lemma 2.13. (|6]J Let A0, Ai,..., Ak-i, F ^ 0 are meromorphic functions of a finite order. If f is a meromorphic solution with p ( f) = of the equation

f( k) + Ak-if(k-i) + ••• + Aif' + Aaf = F,

then

X(f) = X (f) = p(f) = +^.

3. Proof of Theorem 1.5

First, we prove that each meromorphic solution f (^ 0) of the equation (1.4) is transcendental of order p ( f) ^ n. We assume that f (^ 0) is a meromorphic solution of equation (1.4) with p ( f) < n. We can rewrite equation (1.4) as

(Ak-i,i(z)e Pk-1 (z) + Ak-1,2(z)e Qk-l(z)) f(k-i)

+ ... + (yAo,i(z)eP°(z) + Ao,"(z)e^(z)) f = — f(k). ( ' j

Since

max {p (Aji) , j = 0,1,... ,k — 1; i = 1, 2} <n

and

then A3Zf(j), j with

0, 1,

k — 1;i

P( f) < n,

1, ^d f(k) are meromorphie functions of a finite order

p (Ajif(j)) < n and

We have also a0,n = b0,n and a,j,n = Cja0,n, bj,n

p UVVJ < n.

Cj bo,n, Cj > 1, j

1,

1

j,n

= bj,n and therefore deg ( Pj — Po) = deg (Qj — Q0) = n. Since A0,i(z)f = 0, A0,2(z)f = 0, bv Lemma 2,1, we find that the order of growth of the left side of equation (3,1) is n, this contradicts the inequality p ( f(k^ < n. Thus, each meromorphie solution f (^ 0) of equation (1,4) is transcendental with order p (/) 2 n.

iB

rew, ao,n

Kn\ e ^, bo,n = \bo,n\ eld2, 9i, 02 G [0, 2tt). Then

S ( Po, 9) = \ao,n\cos (n9 + 0i) ,6 (Qo, 0) = \bo,n\cos (n0 + .

Since a

j,n

CjOo.

j

n j,n

Cj bo,n, Cj > 1, j

1,

k — 1, and Cj are distinct numbers, we have

5 ( P3, d) = c35 ( Po, d), 5 (Qj, d) = Cj5 (Qo, 9),

(3.2) lave

(3.3)

and there exists exactly one cs such that cs = max [ Cj ,j = 0,1,... ,k — 1} .Let co = 1.

i = 2 i = 2

Case 1. As 9i = 62, because of ao,n = bo,n, we suppose \aonn\ < \bo,n\ without loss of

bounded multiplicity. From (1.4), we have \As,i(z)e p°(z) +As,2(z)e Q°(z)\

ki

<

f ( f(k)

f(s) I f

+

k-i

Z I

■i — n ■i-Aa

A3A(z)e (z) +AJi2(z)e (z)\

f(j)

))

(3.4)

j=o,j=s

Since f is transcendental, then by Lemma 2.2, for a = 2, there exist a set Ei C (1, with mt(Ei) < and a constant B > 0 such that for all z satisfying \z\ = r G [0,1] U Ei, we have

fU)(z)

/ Dlrji iMk+1 _ 1 O /„ -L „ (3 5)

( )

^B [T (2r, f)]k+1, j = 1, 2,...,k, j = s.

Bv (1.4), it follows that the poles of f can be located only at the poles of Aji(z), j = 0,1,... ,k — 1; i = 1, 2. We observe that the poles of f are of uniformly bounded multiplicity. Hence,

A (1/f) ^ max[p (Aji) , j = 0,1,...,fc — 1;i = 1, 2} < n.

By Hadamard factorization theorem, we know that f can be expressed as f(z) = ^z), where

g(z) and d(z) are entire functions with

A (d)=p (d) = X (1/f) <n ^ p(f) = p(g).

For each sufficiently large \z\ = r^t zr = rez&r be a point satisfying \g (zr)\ = M (r, g). By Lemma 2.11, there exist a constant 5r (> 0), a sequence [rm}meM , rm ^ and a set E8 of a finite logarithmic measure such that the estimate

f(z)

^ r,

2s

f(s)(z)

holds for all z satisfying \z\ = rm G E8, rm ^ and argz = 9 G [9r — 5r, 9r + 5r], ( ) ( Po , ) > 0

5(Qj, 9) > 5 (Qo, 9) > 0, 5 (Qj, 9) > 5(Pj, 9) > 5(Po, 9) > 0.

1 ( Cs Cj

(3.6)

0 < e < min < -

{K trf) ^ = s }■

there exists a set E2 C [1, of a finite logarithmic measure such that for all z satisfying |z| = r e [0,1] U E2, r ^ and arg z = 9 e [9r — 5r, 9r + 5r] \ H, where

H = [9 e [0; 2tt) :S(Po, 9) = 0,6 (Qo, 9) = 0}

is a finite set, we have

IAs,i(z)ep°(z) +ASt2(z)e> \As>2(z)— \AM(z)

> exp [(1 — e) CsS (Qo, 9) rn} — exp [(1 + e) csS ( Po, 9) rn}

^ 2 exp [(1 — e) CsS (Qo, 9) rn} ,

I Aj,i(z)ep(z) +Ah2(z)eQ'(z)I ^ \A3tl(z)ep(z)\ + \A3,2(z)eQ>(z)\

^ exp [(1 + e) CjS ( Po, 9) rn} + exp [(1 + e) c3S (Qo, 9) rn} (3.8)

^ 2 exp [(1 + e) CjS (Qo, 9) rn} , j = 0,1, 2,...,k — 1, j = s.

Substituting (3.5), (3.6), (3.7), (3.8) into (3.4), for all z satisfying |z| = rm £ [0,1] UEiUE2UE8, vm ^ and arg z = 9 e [9r — 5r, 9r + 5r] \ H we obtain

2 exp [(1 — e) CsS (Qo, 9) r—} ^ t2—(b [T (2rm, f)]k+1

+ B

k+1 k-1

T (2rm, f)'

k-1 \ £ 2 exp [(1+e) CjS (Qo, 9) r— }

=o,j=s )

j=o, j=s k-1

a r—B [T (2rm, f)]k+1 £ exp [(1+e) CjS (Qo, 9) r— }

j=o,j=s

which gives

k 1

exp [(1 — e) CsS (Qo, 9) r—} ^ 8r—B [T (2rm, /)]k+1 J] exp [(1+e) CjS (Qo, 9) r—} . (3-9)

j=o,j=s

Since 0 < e < min j 1 (^r—r^) , j = s | , then by Lemma 2.6 and (3.9) we obtain

log T (r f) p ( f) = lim sup---—— =

rm^+<x, log T—

and

v loglogT (rm, f) p2 (j) = lim sup--- ^ n.

rm^+<x, log T—

In addition, by Lemma 2.7 and from equation (1.4), we have p2 ( f) ^ n, so p2 (/) = n. If S ( Po, 9) < 0, then by (3.2) and (3.3) we have

S(Qj, 9) < 5 (Qo, 9) < S(Po, 9) < 0, S(Pj, 9) < S(Po, 9) < 0.

By Lemma 2.3, for any given 0 < e < 1, there exists a set E2 C [1, of a finite logarithmic measure such that for all z satisfying IzI = r £ [0,1] U E2, r ^ and argz = 9 e [Or — Sr, 9r + Sr] \ H, where H = [9 e [0; 2tt) : S (Po, 9) = 0,5 (Qo, 9) = 0} is a finite set, we get

\ A3A(z)ep(z) + Aj-2(z)e^ \AjA(z)ep(z)\ + ^(z)

^ exp [(1 — e)S ( Pj, 9) rn} + exp [(1 — e) S (Qj, 9) rn} (3.10) ^ 2 exp [(1 — e)S( Po, 9) rn} , j = 0,1, 2,...,k — 1.

By (1,4) we have

1 ^

f k-i

f(k)

( )

\ Aj,i(z)ep>(z) +Aj,2(z)e^(z)\ ] . (3.11)

=o

Substituting (3.5), (3.6) and (3.10) into (3.11), for all z satisfying \z\ = rm G [0,1] UEi UE2UE8, i"m ^ and arg z = 9 G [9r — 6r, 9r + 6r] \ H we obtain

1 ^ r—B [T 2m, /)]k+^£2exp [(1 — e)S (Po, 9) C} j r^B [T (2Tm, f)]k+1 exp [(1 — e)S (Po, 9) n—}

(3.12)

^ 2 kr^B [T (2Tm, f)]k+1 exp [(1 — e) 6 (Po, 9) v—}

which gives

exp [(e — 1) S ( Po, 9) r—} ^ 2 kr—B [T (2rm, f)]k+1. (3.13)

By Lemma 2.6 and (3.13) we obtain

log+ T ( m, ) p (/) = limsup---—— =

rm—+^ log Tm

and

log+ T ( m, ) p2 ( f) = limsup —2-—— 2 n.

t—log Tm

In addition, by Lemma 2.7 and equation (1.4), we have p2 ( f) ^ n, so p2 (/) = n. Case 2 Assume that 9, = 92.

(i) If 5 ( Po, 9) > 0,5 (Qo, 9) < 0, then by (3.3), we get

6(Pj, 9) > 6(Po, 9) > 0, 5(Qj, 9) < 6(Qo, 9) < 0,

by Lemma 2.3, for any given 0 < e < min[2 (^r-r^ , j = s}, there exists a set E2 C [1,

of a finite logarithmic measure such that for all z satisfying \z\ = r G [0,1] U E2, r ^ and argz = 9 G [9r — 6r, 9r + 6r] \ Hi, where

Hi = [9 G [0, 2tt) :6(Po, 9) = 0,6 (Qo, 9) = 0,6 (Po, 9) = 5 (Qo, 9)}

is a finite set, we have

\Am(z)ep°(z)+As,2(z)eQ°(z)\ 2 \ As>i(z)ep°(z) \ — \ As,2(z)eQ'(z) \

2 exp [(1 — e) CsS ( Po, 9) rn} — exp [(1 — e) c.s6 (Qo, 9) rn}

22 exp [(1 — e) Cs6 ( Po, 9) rn}

(3.14)

\Ajti(z)e (z) +A3,2(z)e * (z)\^ \AjA(z)e p> (z)\ + \AMe

^ exp [(1 + e) Cj6 ( Po, 9) rn} + exp [(1 — e) c36 (Qo, 9) rn} (3.15)

^ 2 exp [(1 + e) Cj6 ( Po, 9) rn} , j = 0,1, 2,...,k — 1, j = s.

By (3.4), (3.5), (3.6), (3.14) and (3.15), for all z satisfying \z\ = r— G [0,1] U Ei U E2 U E8, vm ^ and arg z = 9 G [9r — 6r, 9r + 6r] \ Hi we have

2exp [(1 — £) CsU ( '—)

exp [(1 — e) CsS ( Po, 9) v—} ^r—(B [T (2r—, f)]k+i

k-i x

+ B [T (2T—, f)]k+i Y 2 exp [(1+e) CjS (Po, 9) r—})

j=o,j=s '

k-i

[T (2T—, f)]k+i Y exp [(1 + <0 CjS (Po, 9) r—} j=o,j=s

which gives

k-1

exp {(1 - e) caS ( Po, 9) r—} ^ 8r—B [T (2rm, f)]k+1 ^ exp {(1+e) CjS (Po, 9) r—} . (3.16)

j=o,j=s

Since 0 < e < min j 1 (^r—r^) , j = s | , then bv Lemma 2.6 and (3.16) we obtain

logT (r—, f) p ( f) = lim sup---—— =

rlog f—

and

p2 (f)=limsuplOg}OfT^—lIl

rm^+<x, log T—

In addition, bv Lemma 2.7 and from equation (1.4), we have p2 ( f) ^ n, so p2 (/) = n. If S ( Po, 0) < 0(5 (Qo, 0) > 0, by (3.3), we have

s(Pj,e) < s(Po, 9) < o, s(Qj, o) > s (Qo, 9) > 0.

Bv Lemma 2.3, for any given 0 < £ < min j2 ^, .j = sj, there exists a set E2 C [1,

of a finite logarithmic measure such that for all z satisfying Izl = r £ [0,1] U E2, r ^ and argz = 9 £ [9r — 5r, 9r + 5r] \ Hi, where

H = {9e [0, 2tt) :S(Po, 9) = 0,5 (Qo, 9) = 0,5 (Po, 9) = S (Qo, 9)}

is a finite set, we have

(3.17)

IAsi(z)ep°(z) + A^(z)e| ^ \A8t2(z)eQ°(z)\ — \A8ti(z)ep°(z)\

> exp {(1 — e) csS (Qo, 9) rn} — exp {(1 — e) c.sS ( Po, 9) rn}

> 2 exp {(1 — e) CsS (Qo, 9) rn} ,

I Aj,i(z)ep(z) +Ajt2(z)eQ>(z)I ^ \Ajti(z)ep(z)\ + \Aj^(z)eQ>(z)\

^ exp {(1 + e) CjS (o, 9) rn} + exp {(1 — e) c3S ( Po, 9) rn} (3.18)

^ 2 exp {(1 +e) c^ (Qo, 9) rn} , j = 0,1, 2,...,k — 1, j = s.

Proceeding as in the proof of (i) , for all z satisfying IzI = rm e [0,1] U E1 U E2 U E8, rm ^ and arg z = 9 e [9r — 5r, 9r + 5r] \ H1 we obtain

1

2 —r n- - / - —j ^'mi

exp [(1 — e) c.sS (Qo, 0) v—} ^ C[B [T (2r—, f)]k+i

k-i x

+ B [T (2T—, f)]k+i Y 2exp [(1+e) CjS (Qo, 9) r—}) j=o,j=s '

k-i

^4i%B [T (2T—, f)]k+i Y exp [(1+e) CjS (Qo, 9) r—} ,

j=o,j=s

which gives

exp [(1 — e) CsS (Qo, 9) t— } ^ 8 CB [T (2r—, /)]k+^ exp [(1 + e) Cj5 (Qo, 9) r—} . (3.19)

k-i

j=o,j=s

Since 0 < e < min j i (jr-T^ , j = s} , then by Lemma 2.6 and (3.19) we obtain

log T ( m, ) p ( f) = lim sup —--—— =

log T m

and

p2 (/)= limsup loglogT (rm, ^ 2 n.

rm—+<x, log T m

In addition, by Lemma 2.7 and from equation (1.4), we have p2 (/) ^ n, so p2 (/) = n. (iii) If 8 ( Po, 9) > (Qo, 9) > 0, then bv (3.3), we have

6 ( Pj, 9) > S ( Po, 9) > 0,8 (Qj, 9) > S (Qo, 9) > 0.

We suppose S ( Po, 9) > S(Qo,9) without loss of generality. By Lemma 2.3, for any given

0 < e < min j 2 (^r+^^J , .j = s |, there exists a set C [1, of a finite logarithmic measure

such that for all z satisfying \z\ = r G [0,1]UP2, r ^ argz = 9 G [ 9r — Sr, 9r + Sr ] \ Hi,

where

Hi = [9 G [0, 2tt) :S(Po, 6) = 0,8 (Qo, 9) = 0,8 (Po, 9) = 8 (Qo, 9)} is a finite set, we have

\As,i(z)ep°(z) +As,2(z)eQ°(z)\2 \As>i(z)ep°(z)\ — \As>2(z)eQ°(z)\

2 exp [(1 — e) CsS ( Po, 9) rn} — exp [(1 — e) c.s8 (Qo, 9) rn}

2 2 exp [(1 — e) CsS ( Po, 9) rn}

(3.20)

\Aj,i(z)ep^(z) +Aj,2(z)e*(z)\ ^ \Ajti(z)e(z)\ +

^ exp [(1 + e) CjS ( Po, 9) rn} + exp [(1 + e) c3S (Qo, 9) rn} (3.21)

^ 2 exp [(1 + e) cjS ( Po, 9) rn} , j = 0,1, 2,...,k — 1, j = s.

From (3.4), (3.5), (3.6), (3.20) and (3.21), we have for all z satisfying \z\ = rm G [0,1] U E, U E2 U E8, rm ^ and argz = 9 G [9r — Sr, 9r + Sr] \ Hi

k-i

2 exp [(1 — e) CsS ( Po, 9) r—} ^ 4 r— B [T (2r—, f)]k+i £ exp [(1 + e) CjS (Po, 9) r—} ,

j=o,j=s

which gives

k-i

exp [(1 — e) CsS ( Po, 9) r—} ^ 8 r— B [T (2r—, f)]k+i ^ exp [(1+e) CjS (Po, 9) r—} . (3.22)

j=o,j=s

Since 0 < e < min j2 (, j = then by Lemma 2,6 and (3,22) we obtain

m i- logT (rm, /) p( j) = limsup---=

rm^+<x, log Tm

and

v loglogT (f) p2 (j) = limsup--- ^ n.

rm^+<x, log Tm

In addition, by Lemma 2,7 and from equation (1.4), we have p2 ( f) ^ n, so p2 (/) = n. (iv) If 8 ( Po,9) < 0 5 (Qo,9) < 0, then by (3.3), we have

5 ( Pj, 9) < 8 ( Po, 9) < 0,5 (Qj, 9) < 8 (Qo, 9) < 0.

Let 8 = max ( P0,9) , 8 (Q0, 9)} . Then, by Lemma 2.3, for anv given 0 < £ < 1, there exists a set E2 C [1, + of a finite logarithmic measure such that for all z satisfying Izl = r £ [0,1]UE2, r ^ and argz = 9 £ [9r — 8r, 9r + 8r] \ Hi, where

Hi = {9e [0, 2tt) :8(Po, 9) = 0,8 (Qo, 9) = 0,8 (Po, 9) = 8 (Qo, 9)}

is a finite set, we get

| Aj,i(z)ep(z) +A,2(z)e(z)\ ^\AjA(z)ep(z)\ + \A^(z)e(z)\

^ exp {(1 — £) Cj8 ( Po, 9) rn} + exp {(1 — e) Cj8 (Qo, 9) rn} (3.23) ^2 exp {(1 — £) Cj8 rn} , j = 0,1,...,k — 1.

By (3.5), (3.6), (3.11) and (3.23) for all z satisfying ^ = rm £ [0,1] U Ei U E2 U E8, rm ^ and arg z = 9 £ [9r — 8r, 9r + 8r] \ Hi we have

Jr y/ri^r I ur

' k-1

^3UI m

1 ^CB [T (2rm, /)]k+^2 exp {(1 - e) c,8^}

.3=0 ' k-1

(3.24)

^2 rmkB [T (2rm, /)]k+^ exp {(1 - e) c^^ .

.3=0

Since Cj > 1, j = 1,... ,k — 1 and 8 < 0, we obtain

exp {(1 — e) c38r-m} < exp {(1 — e) 8 r^} , j = 1,...,k — 1

m} { - m

so (3.24) becomes

1 ^ 2 rmkkB [T (2Tm, f)]k+1 exp {(1 — e) 8^} which gives

exp {(e — 1) Si*} ^ 2 rmkBk [T (2rm, f)]k+1. (3.25)

By Lemma 2.6 and (3.25) we obtain

m v logT (^ f)

p(j) = lim sup---=

log Tm

and

y loglogT (f)

P2 (j) = limsup--- ^ n.

rm^+<x, log T m

In addition, by Lemma 2.7 and from equation (1.4), we have p2 ( f) ^ n, so p2 (/) = n. This completes the proof of Theorem 1.5.

4. Proof of Theorem 1.6

(i) Suppose f0 is a meromorphie solution of a finite order to equation (1.5) with poles of uniformly bounded multiplicities. If fi(^ f0) is an another meromorphie solution of a finite order to equation (1.5) with poles of uniformly bounded multiplicities, the function ^ — f0 is a nonzero meromorphie solution to equation (1.4) with p(f1 — f0) < This contradicts

Theorem 1.5. Hence, equation (1.5) has at most one meromorphie solution of a finite order. ( )

bounded multiplicity. By (1.5), it is easy to see that if f has a zero of order a (a > k)dX z0, and B0, B1,..., Bk-1 are analytic at z0, then F must have a zero at z0 of order at least a — k. Hence,

n ^r, —^ ^ kn(^r, + n ^r, —^ + n (r, Bj)

and

N (r, — ^ ^ kN (r, — ^ + N (r, F) + ^N (r, B3) ,

(4.1)

where Bj(z) = Aj1(z)ePj(z) + Aj2(z)eQj(z\ j = 0,1, 2,... ,k — l^w (1.5) can be rewritten as

1 7

— / f(k) f(k-1) fl \ 1 J + Bk-1(z)f—---+ ••• + B1(z)f7 + Bo(z)j

(4.2)

^ f ,-1V / f ■ - > f

By Lemma 2.12 and (4.2), we get for |z| = r outside a set Eg of finite linear measure, we have

7) F) + ^m^r, JJ_^ + m (r, Bj) +

^m ^r, —^ + m (r, Bj) + O (log rT (r, f)).

(4.3)

Therefore, by (4.1), (4.3) and the first main theorem, there holds

1 k-1 —( 1 \ T(r, f) = T(r,-) + O (1) ^ T (r, F) + YT (r, Bj) + kN i r,- j + O (log rT (r, f)) J j=o V J '

for all sufficiently large r </ Eg. For sufficiently large r, we have

O (logrT (r, j)) ^ 2t(r,f).

Let p1 = max {n, p ( F)} . By Lemma 2.4, for any given e > 0, there exists a set E3 c (1, of a finite logarithmic measure such that

(4.4)

(4.5)

T (r, F) ^ rpl +£} T (r, Bj) ^ rpl +£} j = 0,1,...,k — 1,

(4.6)

when |z| = r / [0,1] UE3, r ^ % (4.4), (4.5) and (4.6), for r / [0,1] UE3 UEg sufficiently large, we obtain

1\ 1.

which gives Hence,

and therefore,

T(r, f) ^ rpl+£ + krpl +£ + kN (^r,1 ^ + 1T(r, f)

T(r, f) ^ 2 (k + 1) rpl+£ + 2fcN (V,1 ^ .

P2 ( f) ^ A (f) P2 ( f) ^ A (f) ^ A2 (f).

(4.7)

Since by the definition we have A2 (/) ^ A2 (/) ^ p2 (/), we get

A (/) = A2 (/) = P2 (/) .

On the other hand, max [p (Aji), j = 0,1,... ,k — 1; i = 1, 2} < n and p (Bj) < for all 3 = 0,1,... ,k — f(z) is a solution to (1,5) of infinite order. Hence, by Lemma 2,13

we obtain A (/) = A (/) = p(f) = Since p (Bj) ^ n, by Lemma 2,7, we have

P2 (/) ^ max [n,P (F)} .

(ii) Suppose /0 is a meromorphie solution of the equation (1,5) with finite order, by Lemma 2,12,

Ai)' '0

/0

set Eg of finite linear measure, we have

/ / ) \

we have m r, /b = O (log r), j = 1,... ,k — 1. Using (4.2), we can get for Izl = r outside a

f 1\ f 1\ k ( I(j)\ k-1

m \ , 1) — j + ^^m I r, I + ^^m (r, Bj) + O (1)

V hj V J j=i \ J° J j=o ^m ^r, — ^ + k^m ( r, Bj) + O (log r)

(4.8)

and

j=o

k-1

N (r, jj ^ kN (r, j^J + N (r, + ^N (r, B3). (4.9)

T(r, fo) = T(r, -1) + O (1) ^T (r, + ^T (r, Bj) + kN (r, ^ + O (log r). (4.10) T(r, fo) ^ (k + 1) rpl+£ + kN (^r, jj + O (log r).

By (4.8) and (4.9), we get

-) + O (1) <T fr. —\ +

fo

By (4.6) and (4.10), we get

.^i 1 fo

Hence, we obtain

p( fo) ^ max{ A (fo) ,pi} = max{n,A (fo) ,p (F)} .

()

order p ( f) ^ n. We assume that f is a meromorphie solution to equation (1.5) with p (/) < n. We can rewrite equation (1.5) as

(Ak-M(z)ep*-l(z) + Ak-h2(z)eQ*-l(z)) f(k-1) + ■ ■ ■ + (AoA(z)ePo(z) + A^^z)eQo(z)) f = B(z),

(4.11)

where

B (z) = F (z) — f(k).

Since max [p (Aji ),j = 0,1,...,k — 1; i = 1, 2,p (F)} < n and p(f) < n, then Aji f(j\ j = 0,1,...,k — 1, i = 1, 2, and B(z) are meromorphie functions of a finite order with p {Ajif(j^ < n and p (B) < n. We also have ao,n = bo,n and aj,n = Cjao,n, bj,n = Cjbo,n, Cj > 1, j = 1,... ,k — 1,. Hence, aj,n = bj,n and deg (Pj — Po) = deg (Qj — Qo) = n. Since Ao>1(z)f ^ 0, Ao,2(z)f ^ 0, by Lemma 2.1 we find that the order of growth of the left hand side

n ( B) < n.

solution f to equation (1.5) is transcendental and is of order p( f) ^ n. Let z = r eie, aon = laoJ ei0i, bon = | bonl ei02, 91, Q2 G [0, 2n). Then

5 ( Po, 0) = lao,nl cos (n0 + 61) ,8 (Qo, 9) = |bo,nl cos (n9 + . (4.12)

Since aj,n = Cj ao,n, bj,n = Cj bo,n, Cj > 1, j = 1,... ,k — 1, and Cj are distinct numbers, we have

8 ( Pj, 9) = Cj8 ( Po, 9), 8 (Qj, d) = CjS (Qo, 9), (4.13)

and there exists exactly one cs such that cs = max {Cj ,j = 0, l,...,k — 1} Let cq = 1, ¿i = max {£ ( P0, 9), 8 (Q0, 9)} . We split our proof into two cases: Case 1. Assume that ^ > 0. By Lemma 2,3, for any given

n i 1 (Cs — Cj

0 < £ < mm < n — p1,- -

2 \ cs + Cj

3 = s

)

there exists a set E2 C [1, of a finite logarithmic measure such that for all z satisfying |z| = r e [0,1] U E2, r ^ and arg z = 9 e [9r — 8r, 9r + 8r] \ H3, where

H3 = {9e [0, 2tT) :8(po, 9) = 8 (Qo, 9)}

is a finite set, we have |AM(z)ep°(z) + As,2(z)eQ°(z)l 2\As,i(z)ep°(z)\ — \As,2(z)eQ°(z)\

2 exp {(1 — e) cs8 ( Po, 9) rn} — exp {(1 — e) cs8 (Qo, 9) rn} (414)

22 exp {(1 — e) cs8irn} ,

\ A3A(z)ep>(z) +Ah2(z)eQ>(z)\ ^ \Ajti(z)ep>(z)\ + \AJ;2(z)e(z)\

^ exp {(1 + e) Cj8 ( Po, 9) rn} + exp {(1 + e) c38 (Qo, 9) rn} (4.15) ^2 exp {(1+e) Cj Si rn} , j = 0,1,...,k — 1, j = s.

By (1,5) we have

|AM(z)e + as,2(z) e Q°(z)l

<

f(s)

{

F (z)

+

fk)

k-i ,

+ £ \ Aj,i (z)e p>(z) + Aj,2(z) e *(

j=0,j=s ^

Since f is transcendental, from Lemma 2,2, there exists a set Ei C (1, with mi(Ei) < and constant B > 0, such that for all z satisfying |z| = r e Ei, we have (3,5) holds and by Lemma 2.11, there exists a set E8 of finite logarithmic measure such that |z| = r e E8, lg(z)| = M (r, g) and for r sufficiently large inequality (3,6) holds. We know that f is transcendental with p ( f) 2 n and by the assumptions, the poles of f are of uniformly bounded

multiplicities. By Hadamard factorization theorem, we can express f as f(z) = where g(z) ( )

1'

f(J)

(4.16)

A (d)=p (d) = XyJj<n, p (g) = p (/) 2n.

Let pi = max {p (F) ,p (d)} < n. Since lg(z)| = M (r, g) 2 1, then, by Lemma 2.4 we obtain

1 ^IF^ ^ exp (r^+£) exp (r^ +) = exp (2r* +) (4.17)

as |z| = r e [0,1] U E3, r ^

By (3.5), (3.6), (4.14),(4.15), (4.16) and (4.17), for all z satisfying |z| = rm e e [0,1] UEi UE3 U E8, rm ^ lg(z)| = M(rm, g) and argz = 9 e [9r — 8r, 9r + 8r] \ H3, we have 1

F (z) d(z)F (z)

f(z) 9(z)

exp {(1 — e) Cs8irnm} ^exp (2r£+£) + B [T (2rm, f)]k+i

k-i x

+ B [T (2rm, f)]k+i Y 2exp {(1 + e) c^C}

j=0,j=s )

k-i

exp(2r£+£)B [T (2rm, f)]k+i ^ exp {(1 + e) ^Sirnm}

j=0,j=s

which gives

k-1

exp {(1 - e) cs8-C} ^ 8r^ exp (2 +) B [T (2rm, f)]k+1 £ exp {(1 + e) Cj8- C} . (4.18)

j=0,j=s

Since e < min jn — pi, - (^f-^) , j = s| is arbitrary, so by Lemma 2.6 and (4.18) we obtain

m v log+T (r^ f) p(j) = lim sup---=

rm^+<x, log Tm

and

v loglogT (rm, f) P2 (j) = lim sup--- ^ n.

rm^+<x, log Tm

In addition, by Lemma 2.7 and equation (1.5), we have p2 ( f) ^ n, so p2 (/) = n. Then, each meromorphie solution to (1.5) with poles of uniformly bounded multiplicities is of infinite order and satisfies p2 ( f) = n.

Case 2. Assume that < 0. By Lemma 2.3, for any given £ > 0 we obtain

IA3-i( z)ep(z) +AJi2(z)eQ(z)| ^\AjA(z)ep>+ \Ajt2(z)eQ

^ exp {(1 — e) Cjö ( Po, 9) rn} + exp {(1 — e) c38 (Qo, 9) rn} (4.19) ^2 exp {(1 — £) Cj 5irn} , j = 0,1, 2,...,k — 1.

By (1.5) we get

i [ F (z)

f(k) I f(z)

£ {I A3,1(z)ePi(z) + A3,2(z)e*(z)\ ^ } j . (4.20)

As in Case 1, by (3.5), (3.6), (4.17),(4.19) and (4.20), for all z satisfying ^ = rm G [0,1] U Ex U E3 U E8, rm ^ at which lg(z) | = M(rm, g), and arg z = 9 G [9r — 8r, 9r + 8r] \ H3, we have

1 ^ ^exp (2 +) + B [T (2 rm, f)]k+1 2 exp [(1 — e) c381C} j . (4.21)

Since Cj ^ 1, j = 0,... ,k — 1, rm > R1 > 1 and ^ < 0, we obtain

exp [(1 — e) c3 81 rnm} ^ exp [(1 — e) 81 rnm} , j = 0,...,k — 1

so (4.21) becomes

1 ^ 2 r2k ( k + 1) exp +) B [T (2rm, f)]k+1 exp [(1 — e) 81 r^}

which gives

exp {(e — 1) 61 rnm — +} ^ 2r2k ( k + 1) B [T (2rm, f)]k+1. (4.22)

By Lemma 2.6 and (4.22) we obtain

m v logT (r^ f) p( j) = lim sup---=

log rm

and

y loglogT (rm, f)

P2 (j) = limsup--- ^ n.

rm^+<x, log Tm

In addition, by Lemma 2.7 and equation (1.5) we get p2 (/) ^ n and hence, p2 (/) = n. Then, each meromorphie solution to (1.5) with poles of uniformly bounded multiplicities is of infinite order and satisfies p2 ( f) = n.

Acknowledgements The authors are grateful to the referee for his/her careful reading of this paper.

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Mansouria Saidani,

Department of Mathematics,

Laboratory of Pure and Applied Mathematics,

University of Mostaganem (I'MAB).

В. P. 227 Mostaganem-(Algeria).

E-mail: saidaniman@yahoo. fx

Benharrat Belai'di,

Department of Mathematics,

Laboratory of Pure and Applied Mathematics,

University of Mostaganem (UMAB),

B. P. 227 Mostaganem-(Algeria).

E-mail: benharrat .belaidi@univ-mosta.dz