CC BY
32URAL MATHEMATICAL JOURNAL, Vol. 6, No. 1, 2020, pp. 95-113
DOI: 10.15826/umj.2020.1.008
GROWTH OF ^-ORDER SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS ON THE COMPLEX PLANE
Mohamed Abdelhak Kara, Benharrat Belaïdi1
Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (UMAB), B. P. 227 Mostaganem, Algeria
1benharrat.belaidi@univ-mosta.dz
Abstract: In this paper, we study the growth of solutions of higher order linear differential equations with meromorphic coefficients of <-order on the complex plane. By considering the concepts of <-order and <-type, we will extend and improve many previous results due to Chyzhykov—Semochko, Belaidi, Cao—Xu—Chen, Kinnunen.
Keywords: Linear differential equations, Entire function, Meromorphic function, <-order, <-type.
1. Introduction
Let us consider the following linear differential equations
f(k) + Afc_i(z)/(fc-1) + ■ ■ ■ + Ao(z)f = 0, (1.1)
f(k) + Afc_i(z)f(k-1) + ■ ■ ■ + Ao(z)f = F (z), (1.2)
where k > 2,A0 ^ 0 and F ^ 0. It is well-known that if the coefficients A0, A1,..., Ak-1 and F are entire functions, then all solutions of (1.1) and (1.2) are entire. The equation (1.1) has at least one solution of infinite order if some of coefficients are transcendental. For more details about the growth of solutions of equations (1.1) and (1.2), the reader can refer to [14]. In this paper, we use the standard notations of Nevanlinna value distribution theory of meromorphic functions (see [10, 14, 18, 22]). The term meromorphic function throughout this paper means meromorphic in the whole complex plane C. This will not be recalled in the next statements.
To study the growth of meromorphic functions, we recall the following definitions. For all r € R, we define exp1 r = exp r = er and expp+1 r = exp(expp r), p € N = {1,2,... }. Inductively, for all r € (0, large enough, we define log1 r = log r and logp+1 r = log(logp r), p € N. We also denote exp0 r = r = log0 r, exp_1 r = log1 r and log_1 r = exp1 r.
Definition 1 [13]. The iterated p-order of a meromorphic function f is defined by
m logpT(r,f) _
pp(f) := lim sup-:-, p € N,
log r
where T(r, f) is the Nevanlinna characteristic function of f. If f is an entire function, then the iterated p-order is defined as
~m r logp+i M(r, f) PpU) ■■= limsup-r—-= PpU),
r—log '
where M(r, f) = max{|f (z)| : |z| = r} is the maximum modulus of f.
Note that pi(/) = p(/) is the usual order and p2(/ ) is the hyper-order.
Definition 2 [13]. The growth index of the iterated p-order of a meromorphic function / is defined by
{0 if / is rational,
min {j € N : pj (/) < +œ} if / is transcendental andpj (/) < +œ for some j € N, +œ if pj (/) = +œ for all j € N.
Historically, Bernal [4] was the first one who introduced the idea of the iterated order to study the growth of solutions of complex differential equations. In [13], Kinnunen considered the growth of solutions of equations (1.1) and (1.2) with entire coefficients of a finite iterated p-order and extended many previous results obtained for the usual order and the hyper-order.
Theorem A [13]. Let A0 (z),..., Ak-1 (z) be entire functions such that i (A0) = p (0<p<œ). If either max{i (Aj): j = 1,2,..., k — 1} < p or max{pp (Aj): j = 1,2,..., k — 1} < pp (A0), then every solution / ^ 0 of equation (1.1) satisfies i (/) = p + 1 and pp+1 (/) = pp (A0).
In [3], the second author has extended Theorem A when most of the coefficients A0 (z),..., Ak-1 (z) have the same order by using the concept of iterated p-type as follows.
Theorem B [3]. Let A0 (z),..., Ak-1 (z) be entire functions, and let i (A0) = p (0 < p < œ). Assume that
and where
max{pp (Aj) : j = 1, 2,... , k — 1} < pp (A0) = p (0 < p < +œ)
max{rp (Aj) : pp (Aj) = pp (A0)} < Tp (A0) = t (0 < t < +œ),
logpM(r, /) rp(/) = hmsup rM/) .
Then, every solution / ^ 0 of equation (1.1) satisfies i (/) = p + 1 and pp+1 (/) = pp (A0) = p.
In [5], Cao-Xu-Chen improved Theorems A and B by considering meromorphic coefficients instead of entire coefficients. In [16], Liu-Tu-Shi made a small modification in the original definition of [p, q]-order introduced by Juneja-Kapoor-Bajpai [11] in order to study the growth of entire solutions of equations (1.1) and (1.2). After that, Li and Cao [15] investigated the growth of meromorphic solutions of equations (1.1) and (1.2) with meromorphic coefficients of [p, q]-order which improved many results in [3, 5, 13, 16].
Definition 3 [15, 16]. Let p > q > 1 be integers. The [p, q]-order of transcendental meromorphic function f is defined by
p [p,q](/) = limsup
logp T (r,/ )
logq r
If f is transcendental entire function, then
log p+1M(r,/) P\p,q]U) = hm sup-——-.
Note that P[p,i](f) = Pp(f) is the iterated p-order (see [13, 14]).
Definition 4 [15]. The [p, q]-type of a meromorphic function f with [p, q]-order P[p,q] (f) € (0, is defined by
r log P-iT(r,f)
Definition 5 [15]. Let p > q > 1 be integers. The [p, q]-convergence exponent of the sequence of zeros of a meromorphic function f is defined by
V«](f) = lim sup
logp N (r, 1/f )
r—logq r
where N (r, 1/f) is the integrated counting function of zeros of f in {z : |z| < r} . Similarly, the [p, q]-convergence exponent of the sequence of distinct zeros of f is defined by
V«] (f ) = lim sup
logp N (r, 1//)
r—logq r
where N (r, 1/f) is the integrated counting function of distinct zeros of f in {z : |z| < r}.
Here, we give two results due to Li-Cao in [15] concerning the growth of meromorphic solutions of equations (1.1) and (1.2) when the coefficients are meromorphic functions of [p, q]-order.
Theorem C [15]. Let A0, A1,..., Ak-1 be meromorphic functions such that
max AM (J-^ : j = 1,..., k - 1 j < pM(A0) < +oo.
Then every meromorphic solution f ^ 0 whose poles are of uniformly bounded multiplicities of equation (1.1) satisfies P[p+1;q](f) = p[p,q](A0).
If there exist some other coefficients Aj(j = 1,... ,k — 1) having the same [p,q]-order as A0, then we have the following result.
Theorem D [15]. Let A0,A1 , ...,Ak-1 be meromorphic functions such that A[p>q] (1/A0) < P[p,q](Ao) and
max{p[p,q](Aj) : j = 1,..., k — 1} = P[p,q](Ao) <
max{T[p,g] (Aj) : P[p,q] (Aj) = P[p,q](A0) > ^ j = 1, . . . , k — 1} < T[p,q](A0). Then any non-zero meromorphic solution f whose poles are of uniformly bounded multiplicities of (1.1) satisfies p[p+1,q ](f) ](Ao).
It is clear that Theorem C and Theorem D improve respectively Theorem A and Theorem B from entire coefficients of iterated p-order to meromorphic coefficients of [p, q]-order. Recently, Chyzhykov and Semochko [7] showed that both definitions of iterated p-order and [p, q]-order have the disadvantage that they do not cover arbitrary growth (see [7, Example 1.4]). They introduced more general scale to measure the growth of entire solutions of equation (1.1) called the ^-order (see [20]).
Definition 6 [7]. Let ^ be an increasing unbounded function on [1, . The <p-orders of a meromorphic function f are defined by
.«(/) = limsupMpill, ,l(/) = ,imsupm»
^ r^+œ log r ^ r^+œ log r
If f is an entire function, then the <p-orders are defined by
Definition 7 [1]. Let p be an increasing unbounded function on [1, We define the p-types
of a meromorphic function f with p-order € (0, by
e^(eT(r>f)) (r,f))
ty(/) = limsup " o , r(/) = limsup , .
r^+ro rp^(f) r^+ro rp^(f)
If f is an entire function, then the p-types are defined as
0 e^(M (r,f)) i e^(log M (r,f))
Tv(f) = limsup—o . , f (/) = limsup--.
r^+ro r^V^) r^+ro r^V")
By symbol $ we define the class of positive unbounded increasing functions on [1, , such that
p(ect)
(p(et) grows slowly i. e., Vc > 0 : lim —j-jj = 1.
Example 1. Let f be a meromorphic function. One can see that p(r) = logp r, (p > 2) belongs to the class $ and p(r) = log r / $. Moreover, the p^(f) order of the function f coincides with its iterated p-order, i. e., pj,(f) = pp(f). As a particular case, for p = log2 € $ we have P0og (f) = P1(f) and p1og (f) = p2(f) which are respectively the usual order and the hyper-order of f.
The following result due to Chyzhykov-Semochko [7] investigates the growth of entire solutions of equation (1.1) when the coefficients are entire functions of p-order.
Theorem E [7]. Let p € $ and Ao, A1,..., Ak-1 be entire functions such that
max{p° (Aj), j = 1,..., k - 1} < p° (Ao). Then every solution f ^ 0 of (1.1) satisfies pj,(f) = p° (A0).
We recall that the linear measure of a set E C (0, is defined by
r+ro
m(E) = xe(t) dt
o
and the logarithmic measure of a set F C (1, is defined by
1t
where xa is the characteristic function of a set A. The upper density of a set E C (0, is defined by
densE = lim sup"7^ n rD _
r^+ro r
The upper logarithmic density of a set F C (1, is defined by
1—,--lrn(F Pi [1, r])
log dens I = lim sup---.
r^+ro log r
Definition 8 [10, 22]. For a € C = Cu{oo}, the deficiency of a with respect to a meromorphic function f is defined as
. fm(r, 1/(/ - a)) N(r, !/(/ - a))
b a, / = limmf-——--= 1 - limsup-—-—--, a A oo,
r^+ro T(r, f) r^+ro T (r, f)
xf f\ v • tm (r,f) , r N (r,f) b oo,/ = lim mf = 1 - lim sup
r^+ro T (r, f) r^+ro T (r, f)
Recently, the second author has studied the growth of entire solutions of equation (1.1) when the coefficients are entire functions of p-order and obtained the following results.
Theorem F [2]. Let G be a set of complex numbers z satisfying log dens {\z\ : z € G} > 0. Let p € $ and let A0, A1,..., Ak-1 be entire functions satisfying
max{p° (Aj) : j = 0,1,... , k — 1} < a (0 < a <
Suppose, there exists a real number ft satisfies 0 < ft < a such that for any given e (0 < 2e < a — ft), we have
T(r, A0) > log (p-1((a — e) log r))
T(r, Aj) < log (p-1 (ft log r)) , j = 1,..., k — 1 as |z| ^ for z € G. Then every non-zero solution f of equation (1.1) satisfies pj,(f) = a.
Theorem G [1]. Let A0 (z),..., Ak-1 (z) be entire functions, and let p € Assume that max{p° (Aj) : j = 1,..., k — 1} < p° (Ao) = p < (0 < p <
and
max{f° (Aj) : p° (Aj) = p° (Ao)} < f° (Ao) = t (0 < t < . Then every solution f ^ 0 of (1.1) satisfies p^ (f) = p° (A0).
2. Main results
The aim of this paper is to investigate the growth of meromorphic solutions of equations (1.1) and (1.2) with meromorphic coefficients of finite p-order. By using the concept of p-order, we can cover arbitrary growth of solutions of equations (1.1) and (1.2) which improves several results in [1-3, 5, 7, 13]. To do that, we firstly introduce the following quantities by an analogous manner with the definitions of the p-orders.
Definition 9. Let p be an increasing unbounded function on [1, We define the p-
convergence exponents of the sequence of zeros of a meromorphic function f by
r^+œ log i r^+œ log I
Similarly, the notations A° (/ ) and A^(/) can be used to denote the (p-convergence exponents of the sequence of distinct zeros of /.
Now, we list our main results.
Theorem 1. Let p € $ and Ao, Ai,..., Ak-i be meromorphic functions. Suppose, there exists one coefficient As (s € {0,1,..., k — 1}) such that
max |p° (Aj), A° (^j :./ 0.1.....k 1 (j ^ s) j < p° (A,) < +oo.
Then every transcendental meromorphic solution / whose poles are of uniformly bounded multiplicities of (1.1) satisfies
P^(/) < P° (As) < p° (/). Furthermore, if all solutions of (1.1) are meromorphic solutions, then there is at least one mero-
?^(/i) = P° (
morphic solution, say /i, verifies pi(/i) = p^(A0).
Remark 1. By setting p (r) = logp+1 r (p > 1) in Theorem 1, we obtain Theorem 2.2 in [5]. Theorem 2. Let p € $ and Ao, A1,..., Ak-1 be meromorphic functions such that
max |à° , (aj) :./ 1.....k l| < p%(a0) < +oo.
Then every non-zero meromorphic solution f whose poles are of uniformly bounded multiplicities of (1.1) satisfies pj(f) = p° (Ao).
Remark 2. Clearly, Theorem 2 is an extension of Theorem E from entire solutions of equation (1.1) to the case of meromorphic solutions of equation (1.1) with meromorphic coefficients instead of entire coefficients. Furthermore, by setting p (r) = logp+1 r (p > 1) in Theorem 2, we obtain Theorem A when the coefficients of (1.1) are entire functions.
If there exist some other coefficients Aj (j = 1,..., k — 1) having the same p-order as A0, then we have the following result.
Theorem 3. Let p € $ and A0,A1 ,...,Ak-1 be meromorphic functions such that A° (1/Ao) < p° (Ao) and,
max{p° (Aj) : j = 1,..., k - 1} < p° (Ao) = po < (2.1)
max{r°(Aj) : p° (Aj) = p° (Ao) > 0, j = 1,..., k - 1} < r°(Ao) = To (0 < To < . (2.2)
Then any non-zero meromorphic solution f whose poles are of uniformly bounded multiplicities of (1.1) satisfies p* (f) = p° (Ao).
Remark 3. Namely, Theorem 3 extends Theorem G from entire solutions of equation (1.1) to meromorphic solutions. Furthermore, by setting p(r) = logp+1 r (p > 1) in Theorem 3, we obtain Theorem 2.1 in [5] and Theorem B when the coefficients of (1.1) are entire functions.
Theorem 4. Let p € $ and A0, A1,..., Ak-1, F ^ 0 be meromorphic functions such that A° (1/Ao) < p° (Ao) and,
max {pj (F ),p° (Aj ) : j = 1,...,k — 1} < p° (Ao) < (2.3)
Then every meromorphic solution f whose poles are of uniformly bounded multiplicities of (1.2) satisfies
(f ) = Aj (f ) = pjf ) = p° (Ao) with at most one exceptional solution fo satisfying pj(fo) < p° (Ao).
Remark 4■ Theorem 4 is a counterpart of Theorem 1.6 in [15]. Moreover, if we choose p (r) = logp+1 r (p > 1) in Theorem 4, then we obtain a special case of Theorem 2.6 in [21].
Theorem 5. Let p € $ and Ao, A1,..., Ak-1, F ^ 0 be meromorphic functions such that
max{p° (Aj) : j = 0,..., k — 1} < pj(F).
If all solutions f of (1.2) are meromorphic functions whose poles are of uniformly bounded multiplicities, then there holds pj(f) = pj(F) for all solutions of (1.2).
Remark 5. Theorem 5 is a counterpart of Theorem 1.7 in [15]. Furthermore, if we choose p (r) = logp+1 r (p > 1) in Theorem 5, then we obtain a special case in [13, Remark 4.1, p. 399] when the coefficients of equation (1.1) are entire functions.
Theorem 6. Let G C (1, be a set of complex numbers z satisfying
logdens{|z| : z € G} > 0.
Let p € $ and Ao, Ai,..., Ak-1 be meromorphic functions satisfying 5 (to, Ao) = 5 > 0 and
max{p° (Aj) : j = 0,1,... , k - 1} < a (0 < a < +to).
Suppose, there exists a real number ft satisfies 0 < ft < a such that for any given e (0 < 2e < a - ft), we have
T(r, Ao) > log (p-1((a - e) log r)) (2.4)
and
T(r, Aj) < log (p-1(ft log r)), j = 1,..., k - 1 (2.5)
as |z| = r ^ +to for z € G. Then every non-zero meromorphic solution of equation (1.1) satisfies pj,(f) = a.
Remark 6. Theorem 6 extends Theorem F from entire solutions of equation (1.1) to meromorphic solutions.
Theorem 7. Let G C (1, +to) be a set of complex numbers z satisfying
log dens{|z| : z € G} > 0.
Let p € $ and Ao, A1,..., Ak-1, F ^ 0 be meromorphic functions satisfying
max{p° (Aj) : j = 0,1,..., k - 1} < a (0 < a < +to).
Suppose, there exists a real number ft satisfies 0 < ft < a such that for any given e (0 < 2e < a - ft), we have
|Ao(z)| > p-1((a - e) log r) (2.6)
and
|Aj(z)| < p-1(ftlogr), j = 1,...,k - 1 (2.7)
as |z| = r ^ +to for z € G. Then, the following conclusions hold
(i) If p\p(F) > a, then all meromorphic solutions f whose poles are of uniformly bounded multiplicities of equation (1.2) satisfy pj,(f) = p^(F).
(ii) If p^(F) < a, then every meromorphic solution f whose poles are of uniformly bounded, multiplicities of (1.2) satisfies
A^ (f) = AJ (f) = pjf) = a
with at most one exceptional solution fo satisfying p^(fo) < a.
Remark 7. Clearly, Theorem 7 is an improvement of Theorem 1.15 in [2] from entire solutions of equation (1.2) to meromorphic solutions. Furthermore, Theorem 7 is a counterpart of Theorem 1.8 in [15].
3. Preliminary lemmas Proposition 1 [7]. If p € $, then
Vm > 0, Vfc > 0 : ^ ^logs™) +qo x +qo ^ ^
V5 > 0 : ^^Hd ^» +QO ^ +QO_ (3.2)
log p-1 (x)
Remark 8 [7]. We can see that (3.2) implies that
Vc > 0, p(ct) < p(tc) < (1 + o(1))p(t), t ^ (3.3)
Proposition 2 [7]. Let p € $ and f be an entire function. Then
p£(f ) = pf), j = 0,1.
Lemma 1 [6]. Let f be a meromorphic solution of equation (1.1), suppose that not all coeffi-
fc_1
cients Aj are constants. Given a real number 7 > 1, and denoting T(r) = ^ T(r, Aj), then the
j=0
inequalities
log m(r,f) <T(r){(log r) logT(r)}7 if s = 0, logm(r,f) < r2s+Y-1T(r){logT(r)}Y if s> 0
take place outside of an exceptional set with J ts-1 dt <
Es
Lemma 2 [8]. Let f1,f2,...,fk be linearly independent meromorphic solutions of equation (1.1) with meromorphic coefficients A0, A1,..., Ak-1. Then
m(r, Aj) = of log ( max T (r, f ))), j = 0,1,...,k — 1. V 1<i<fc /
Lemma 3 [9]. Let f be a transcendental meromorphic function and let a > 1 be a given constant. Then, there exists a set E1 c (1, with finite logarithmic measure and a constant Ba > 0 that depends only on a and i, j (j > i > 0) such that for all z satisfying |z| = r / [0,1] U E1, we have
f(j )(z)
< ^ r(ar'/} (log° r) log T(ar,/)'J 1
f (i)(z)
Lemma 4 [12]. Let f be a meromorphic function and p € $. Then
p£(f') = p£(f) for j = 0,1.
Lemma 5 [7, 12]. Let p € $ and f1, f2 be two meromorphic functions. Then
(i) pf + f2) < max jp^(f1), p^(f2^ and j(f1f2) < max {p^fO, p^(f2)} for j = 0,1.
(ii) If pf < j(f2), then pf + f2) = p^ff = pUf2) for j = 0,1.
Lemma 6. Let p € $ and f be a meromorphic function. Then, for any set E2 C [0, with finite linear measure, there exists a .sequence {rn,rn / E2} such that
Proof. The definition of p"(f) implies that there exists a sequence {sn, n > 1},sn ^ such that
Sn^+M log sn ^
Setting m(E2) = 5 < Then, for rn € [sn, sn + 5 + 1]\E2, we have
<fi(T(rn, /)) > p(T(s„, /)) _ p(T(s„,/))
l0g(S" + * + 1) logSra + logfl + ^V
Hence
By
we deduce that
lim ïEM1> lim -.-m,.../)) ,
log rn / d + lx ^ log s„ + log H--
.....< Ums„Pœ» - J
log r„ r^+œ log r
lim —---- < lim sup—-—-- = p^if),
n
log rn r<f
Similar proof for p" (f). □
Lemma 7. Let p € $ and f be a meromorphic function satisfying 0 < p° (f) < and 0 < T0(f) < Then, for any given n < T0(f), there exists a set E3 C [0, with infinite logarithmic measure such that for all r € E3, we have
p(eT(rf)) > log(nrp£(/)).
Proof. We denote p° (f) = po and t0(f) = to. The definition of t0(f) implies that there exists a sequence {rm, m > 1} tending to satisfying
. 1 . ejP(eT(rm,f))
1H--)rm < rm+1 and lim -T-= r0.
V mj m^+ro rm0
Then, for any given e (0 < e < t0 — n), there exists an integer mi such that for all m > mi, we have
ev(eT(rm'f^ > (to — e)rP0. (3.4)
Since n < t0 — e, there exists an integer m2 such that for all m > m2, we have
"' V ,-0-, (3.5)
^ m + 1 / t0 — e
Taking m > m3 = max{m1,m2}, it follows from (3.4) and (3.5) that for any r € [rm, (1 + 1/m) rm]
P0
> (to - e)r% > (to - e) ( ^ ) >
MeTir'f]) > MeTirm'f)) ' — - ^ I mr
r
n
s
n
s
n
Thus
p(eT(r'f)) > log(nrp^(/)}.
+00
Setting E3 = |J [rm, (1 + 1/m) rm] , then the logarithmic measure lm(E3) of E3 satisfies
m=m3
(1+l/m)rm
+0 r +0 1
'■»№»)=£ / 7= E k*(i+ £) = +«>.
m=m3 r m=m3
' m
□
Lemma 8. Let A0, A1,..., Ak-1, F ^ 0 be meromorphic functions and let f be a meromorphic solution of equation (1.2). If max{ p^ (F ),p^ (Aj) : j = 0,1,...,k — 1} < p^(f), then
A^ (f ) = Aj,(f )= pjf).
Proof. Equation (1.2) can be written as
1 1 / f(k) f(k-1) f' \
y = ^ + Ak~iLY~ +'''+ Alf + A°) • (3-6)
If f has a zero at z0 of order l > k and if A0, A1,..., Afc_1 are all analytic at z0, then F has a zero at z0 of order at least l — k. Then
1 11 ^ 1
" -- 1 11 ( r, — ) + > n(
n(r, — j < fc • ??^r, —j + —j + Aj)
j=0
and
k-i
A(r,-) Ä(r,-) +A(V,-) +J]A(r,Aj). (3.7)
By the lemma of logarithmic derivative [10] and (3.6), we get that
1 7
1 1 k-i
m(r, -7j < m(r, — j + ^ m(r, Aj) + 0(log r + logT(r, /)) (3.8)
j=0
holds for all |z| = r / E4, where E4 is a set of finite linear measure. By (3.7), (3.8) and the Nevanlinna's first main theorem, we obtain
T(r, /) = T(r, y) + 0( 1) = m(r, y) + N(r, y) + 0( 1)
/7 w V/ V'/,
1 k-i
< k -N(r, -) + T(r, F) + Aj) + 0(logr + logT(r, /))
(3.9)
/
holds for all sufficiently large r / E4. We denote
p = max {pj,(F), p^(Aj) (j = 0,1,..., k - 1)} . According to Lemma 6, there exists a sequence {rn, rn / E4} such that
log rn ^
So, if rn / then for any given e (0 < 2e < p1 — p) we get
T(r„,/) > p-1((pi — e) log r„). (3.10)
We have
max |T(r„,F),T(r„, A,)} < p-1((p + e) logr„), (3.11)
j=0,1,...,fc-1
O(log r„ + log T (rn,/ )) = o(T (r„,/ )). (3.12)
Since e (0 < 2e < p1 — p), then from (3.10), (3.11) and Proposition 1, we obtain
T(rn,F) T(rn,Aj) \ exp {log p-1((p + e) log rn)}
m / _ __ \
i=o,i,...,fc-i \ T(rn, /) ' T(r„,/) J - exp {logp_1((Pi — t) logr„)}
= exp {log p-1((p + e) log rn) — log p-1((pi — e) log rn)} (3.13)
(f, log p-1((pi — e) log rn)V .
= «<> U' ~ log v-mn + c) log r.)J 108 V ((" + £) l0g r">I °
as rn ^ By substituting (3.12) and (3.13) into (3.9) we deduce that for sufficiently large rn / E4, there holds
(l-o(l))T(rn,f) <kJV (rn,±) .
From this inequality, by the monotonicity of p and (3.3), we obtain p"(f) < A"(f). In addition, we have by definition that A* (f) < A"(f) < p*(f). Hence A"(f) = A"(f) = p*(f). □
Lemma 9. Let f be a meromorphic function. If p° (f) = p < then p"(f) = 0.
Proof. Suppose that p° (f) = p < Then, for any given e > 0 and sufficiently large r, we have
T(r,f) < log(p-1 ((p + e) log r)). By Karamata's theorem (see [19]), it follows that p(e*) = to(1) as t ^ Hence,
V r^+M log r r^+M log r
(logT(r,/))o(1) (loglog(p-1((p + e) logr)))°(1)
= lim sup-:- < lim sup------= 0.
r^+M log r r^+M log r
□
4. Proofs of the main results
Proof of Theorem 1. (i) We first prove that p" (f) < p° (As) < p° (f) holds for every transcendental meromorphic function satisfying (1.1). From equation (1.1), we know that the poles of f can only occur at the poles of A0, A1,..., Ak-1, note that the multiplicities of poles of f are uniformly bounded, so we have
k-1
N (r,/ ) < CiNV(r, / ) < C^ NV(r,Aj ) < C max{N (r,Aj ) : j = 0,1,...,k — 1} < O(T (r,As)),
j=0
where C and C are two suitable positive constants. Hence
T(r,f) < m(r, f)+ O(T(r,As)).
This inequality and Lemma 1 lead to
T(r, f) < m(r,f) + O(T(r, )) < O(eT(r'As)[(logr)logT(r,As)]Y), 7 > 1
outside of an exceptional set E0 with finite logarithmic measure. By the monotonicity of the function p and (3.3), we obtain p^(f) < p°(As).
On the other hand, equation (1.1) can be written as
f(k) f (k-1) f (s+1) f (s-1) f
T^T + ^^T^r + '-' + ^T^r + ^^T^r + '-' + ^T^)
f / f(k) f(k-1) f(s+1) f(s 1) \
= y^ ( — + + ''' + As+1f~ + As~i:f~ + ' ' ' + J *
By the lemma of logarithmic derivative and the fact that
m (r, < T(r, /) + T (V, yL) = T(r, /) + T(r, /(*>) + O(l) = 0(T(r, /)),
it follows that
T(r, As) < N(r, As) + m(r, Aj) + O(log r + log T(r, f)) + O(T(r, f)) (4.1)
which holds for all |z| = r / E5 where E5 is a set of finite linear measure. By Lemma 6, it follows that there exists a sequence {rn, n > 1}, rn ^ such that for |zn| = rn / E5
^e-)_=o{Aa) =
log rn ^
and so
T(r„,As) > log(p-1((po - e) log rra)). (4.2)
Under the assumption n = max{p° (Aj), (1/As) : j = s} < p° (As) = p0, we have
N(r„, As) < log(p-1((n + e) log r„)), (4.3)
m(r„, Aj) < T(r„, Aj) < log(p-1((n + e) log r„)), j = s (4.4)
provided for any given e that verifies 0 < 2e < p0 — n. Substituting (4.2), (4.3) and (4.4) into (4.1),
we get
(1 — o(1)) log(p-1 ((po — e) log rra)) < O(log r„ + logT(r„, f)) + O(T(r„, f)) = O(T(r„, f)).
Applying (3.3), one can deduce that p°(As) = p0 < p°(f).
(ii) Now, we prove that there exists at least one meromorphic solution that satisfies p^(f) = p° (As). Let {f1, f2,..., fk} be a solution base of equation (1.1). By Lemma 2, we have
em(r'As) < o( max T(r, f)), s€ {1, 2,..., k — 1}. V 1<i<k /
If N(r,As) > m(r,As), so T(r,As) < 2N(r,As), then p%{As) < A° ^J-j . This contradicts our
assumption A° ^^^ < P%(AS) and asserts that N(r, As) < m(r, As). Hence, for sufficiently large r, we have
eT(r'As) = O(em(r'As)) < o( max T(r,f)).
V 1<i<k J
This implies that there exists at least one solution of {f1, f2,..., fk}, say f1, that satisfies eT(r,As) < O(T(r, f1)). By this inequality and (3.3) and the monotonicity of p, we obtain
p° (As) < pj,(fi).
We have proved in the first part that p"(f1) < p"(As). Therefore, p"(f1) = p"(As). □
Proof of Theorem 2. Assume that f is a non-zero meromorphic solution whose poles are of uniformly bounded multiplicities of (1.1). Equation (1.1) can be written as
i f(k) f(k-i) fA
By the lemma of logarithmic derivative and the above equation, we have
k-i k ( f co\
m.(r, A0) < ^ m(r, Aj) + ^ m r, -— +0( 1)
j=i j=i V f /
^^ ml r,
j
k-1
< ^ m(r, A,) + O(log r + log T(r, /)) j=1
(4.5)
holds possibly outside of an exceptional set E6 C (0, with finite linear measure. From this inequality, it follows
T (r, Ao) = m(r, Ao) + N (r, Ao )
k-1
< N(r, A0) + ^ m(r, A,) + O(log r + log T(r, /))
(4.6)
j=1
holds for r / E6. By Lemma 6, it follows that there exists a sequence {rn, n > 1}, rn ^ such that for |zn | = rn / E6
Pie-)_=o{Ao) =
log rn ^
and so
T1(rn, Ao) > log(p-1 ((po — e) log rn)) (4.7) under the assumption n = max{p"(Aj), A" (1/A0) : j = 0} < p"(A0) = p0, we have
N (rn ,Ao) < log(p-1((n + e) log rn)), (4.8)
' m -^j) < T(rn, Aj
m(rn, A,) < T(rn, A,) < log(p-1((n + e) log rn)), j = 0 (4.9)
provided for any given e that verifies 0 < 2e < p0 — n. Substituting (4.7), (4.8) and (4.9) into (4.6),
we get
(1 — o(1)) log(p-1((po — e) log rn)) < O(log rn + log T(rn, f)).
Applying (3.3), one can deduce that p°(A0) = p0 < p^(f).
On the other hand, from Theorem 1, we have p° (A0) > p^(f). We deduce finally that every meromorphic solution f ^ 0 whose poles are of uniformly bounded multiplicities of (1.1) satisfies p^(f) = p° (A0). □
Proof of Theorem 3. Assume that f is a non-zero meromorphic solution whose poles are of uniformly bounded multiplicities of (1.1). If A° (1/A0) < p° (A0) and
max{p° (Aj) : j = 1,... ,k — 1} < p° (Ao) < then by Theorem 2, we obtain pj, (f) = p° (A0). Suppose that A° (1/A0) < p° (A0) and
max{p° (Aj) : j = 1,..., k — 1} = p° (A0) = p0 (0 < p0 < , max{rQ (Aj) : p° (Aj) = p° (A0)} < rQ (A0) = T0 (0 < tq < .
Then, there exists a set J C {1,..., k — 1} such that p° (Aj) = p° (A0) = p0 (j € J) and rQ (Aj) < rQ (A0) = t0 (j € J). Hence, there exist two constants ft and ft such that
max{rQ(Aj) : j € J} < ft < ft < rQ(Ao) = ro. The definition of the type rQ(Aj) implies that for r sufficiently large
em(r'A) < eT(r'A) < p-1 (log(ftrP0)), j € J (4.10)
and
em(r'A) < eT(r'A) < p-1(log(rp0)) < p-1(log(ft rp0)), j € {1,..., k — 1} \ J, (4.11)
where 0 < pQ < p0. Since A0 = A° (1/A0) < p° (A0) = p0, then for any given e (0 < 2e < p0 — A0) and sufficiently large r, we have
eN(r'Ao) < p-1(log(rA°+£)) < p-1(log(rp0-£)) < p-1 (log(ftrp0)). (4.12)
By Lemma 7, there exists a set E3 c [1, with infinite logarithmic measure such that for all r € E3, we have
eT(r'A0) > p-1 (log(ftrp°)). (4.13)
By substituting (4.10), (4.11), (4.12) and (4.13) into (4.6), we obtain
(1 — o(1))log(p-1 [log(ft rP0)]) < O(log r + log T(r, f)) (4.14)
for all r € E3\E6. Since E3\E6 is a set of infinite logarithmic measure, then there exists a sequence of points |zn| = rn € E3\E6 tending to Hence, by (4.14) we have
(1 — o(1)) log(p-1[log(ft2rn0)]) < O(log r„ + log T(rra, f))
holds for all satisfying |zn| = rn € E3\E6 as |zn | = rn ^ By the monotonicity of
p-1 and (3.3), we obtain p°(A0) < pj,(f). By Theorem 1, we have pj,(f) < p°(A0). Therefore p^(f) = p° (A0) which completes the proof. □
Proof of Theorem 4. Since all solutions of equation (1.2) are meromorphic functions, all solutions of the homogeneous differential equation (1.1) corresponding to equation (1.2) are also
meromorphic functions. We assume that {f1,..., fk} is a meromorphic solution base of (1.1), then any solution of (1.2) has the form
f = C1f1 + C2f2 + ••• + Ck fk, (4.15)
where c1, c2,..., Ck are meromorphic functions satisfying
cj = F ■ Gj (f1,...,fk) ■ W-1(f1,...,fk), j = 1,2,..., k, (4.16)
where Gj(f1,..., fk) are differential polynomials in {f1,..., fk} and their derivatives and W-1(f1,..., fk) is the Wronskian of {f1,..., fk}. We have by Theorem 2
pj(fj ) = p°(Ao), j = 1,..., k.
By Lemma 4, Lemma 5, (4.15) and (4.16), we get
pj(f) < max{pj(fj) ( j = 1,..., k), pj(F)} = pj (Aq).
In order to show that all solutions f of equation (1.2) satisfy pj(f) = pj(A0) with at most one exceptional solution, say f1, satisfying pj(f1) < pj(A0), we suppose that there exist two distinct meromorphic solutions f1 and f2 of equation (1.2) satisfying pj(fj) < pj(A0), i = 1,2. Then, f = f1 — f2 is also a non-zero meromorphic solution of (1.1) and satisfies
pj(f) = pj(f1 — f2) < max{pj(f1),pj(f2)} < pJ(Ao)
which contradicts Theorem 2. By (2.3) for all solutions f of equation (1.2) satisfying pj(f) = pj(A0), by Lemma 9, we have
max{pj(F), pj(Aj) (j = 0,1,..., k — 1)} = pj(F) < pj(Aq) = pj(f).
By Lemma 8, we have Aj(f) = Aj(f) = pj(f) and hence Theorem 4 is proved. □
Proof of Theorem 5. Let f be a meromorphic solution of equation (1.2) and {f1,..., fk} be a meromorphic solution base of (1.1) corresponding to equation (1.2). By a similar discussion as in the proof of Theorem 4, it follows from Lemma 4, Lemma 5, (4.15) and (4.16) that
pj(f) < max{pj(fj) (j = 1,... , k), pj(F)}.
By the first part of the proof of Theorem 1, one can show easily that
pj(fj) < max{pj(Aj) : j = 0,..., k — 1} (4.17)
for j = 1,..., k. We obtain from the assumptions of Theorem 5 that pj(fj) < pj(F) and thus
pj(f) < pj(F).
On the other hand, by Lemma 4, Lemma 5 and a simple order comparison from equation (1.2), we get
pj(F) < max{pj(Aj) (j = 0,..., k — 1), pj(f)}. Since pj(Aj) < pj(Aj) < pj(F) (j = 0,..., k — 1), then
pj(F) < pj(f).
Therefore, p"(f) = p"(F). □
Proof of Theorem 6. Assume that f is a non-zero meromorphic solution whose poles are of uniformly bounded multiplicities of (1.1). Set G\ = {|z| = r : z € G}, since \ogcl,ens{\z\ : z €
f dr
G} > 0, then G\ is a set with / — = +oo. Set
JG i r
5 (oo, A0) = lim inf ™ ^ f = 5 > 0. (4.18)
v ' r^+M T (r,A0) v 7
Thus, for sufficiently large r, we have
m(r,A0)>^6T(r,A0). (4.19)
By substituting (2.4), (2.5) and (4.19) into (4.5), we obtain for sufficiently large r and any given e (0 < 2e < a — ft)
log (v?_1((a - e) log r)) < ^ST (r, A0) < m(r, A0)
k-i k
m I r -
f
< Aj) + J^m ( r> ^T" J + Oi1)
j=i j=i k-i
< ^T(r, Aj) + O(log r + log T(r, f)) j=i
< (k — 1) log (p-1 (ft logr)) + O(logr + logT(r, f)),
it follows that
(1 — o(1)) log (p-1((a — e) log r)) < O(log r + log T(r, f)) (4.20)
holds for all z satisfying |z| = r € G1 \ E6 as |z| = r ^ Since G1 \ E6 is a set of infinite
logarithmic measure, then there exists a sequence of points |zn | = rn € G1 \ E6 tending to Hence, by (4.20) we have
(1 — o(1)) log (p-1((a — e) log rn)) < O(log rn + log T(rn, f))
holds for all zn satisfying |zn| = rn € G1 \ E6 as |zn| = rn ^ By the monotonicity of p-1 and arbitrariness of e (0 < 2e < a — ft), one can obtain p"(f) > a.
On the other hand, it follows by a similar proof as in the first part of Theorem 1 that p"(f) < a. Therefore p"(f) = a. □
P r o o f of Theorem 7. (i) If p"(F) > a, then it follows from Theorem 5 that p "(f) = p" (F). (ii) If p"(F) < a, we prove that p1 = p "(f) = a for any non-zero meromorphic solution whose poles are of uniformly bounded multiplicities of (1.1). We show firstly that p1 = p"(f) > a. Without loss of the generality, we suppose the contrary p1 < ft < a. Set G2 = {|z| = r : z € G},
/dv
— = +oo. From Lemma 3, there exists a set
G2
E1 C (1, with finite logarithmic measure and a constant B > 0 such that for all z satisfying |z| = r / [0,1] U E1, we have
f (j)(z)
f(z)
< B[T(2r,f)]k+1, j = 1,...,k.
(4.21)
If f is a non-zero meromorphic solution of equation (1.1), then
|Ao (z) I <
f(k) (z)
f (z)
+ |Ak-i(z)
f(k-1) (z)
f (z)
+ ••• + |Ai (z)
f ' (z)
f (z)
(4.22)
By the definition of p1 = p^(f) and substituting (2.6), (2.7), (4.21) into (4.22), we obtain p-1((a - e) log r) < |Ao(z)| < k B p-1 (ft log r)[T(2r, f)]fc+1
< k B p-1 (ft log r)
((pi + |)log2r
k+i
(4.23)
<
((/3 + -)log2r
k+2
< p-1((ft + e) log r)
holds for all z satisfying |z| = r € G2 \ ([0,1] U E1) as |z| = r — Since G2 \ E1 is a set of infinite logarithmic measure, then there exists a sequence of points |zn| = rn € G2 \ E1 tending to Hence, by (4.23) we have
p-1((a - e) log r„) < p-1((ft + e) log r„)
holds for all zn satisfying |zn| = rn € G2 \ E1 as |zn| = rn — By the monotonicity of p-1 and arbitrariness of e(0 < 2e < a — ft), one can see that a < ft which contradicts our assumption. Then, pj(f) > a.
On the other hand, it follows by a similar proof in Theorem 1 that
pj(f) < a.
Therefore pj(f) = a. In order to show that all solutions f of equation (1.2) satisfy pj(f) = a with at most one exceptional solution, say f0, satisfying pj(f0) < a, we suppose that there exist two distinct meromorphic solutions f0 and f of equation (1.2) satisfying max {pj(f0),pj(fo)} < a. Then, f = f0 — f^ is also a non-zero meromorphic solution of (1.1) and satisfies
Pi(f) = Pl(f0 — fo) < max {Pl(f0), pi(foH < a
which contradicts the proof of the first part of (ii). By assumptions of Theorem 7, for all solutions f of equation (1.2) satisfying pj(f) = a, we have by Lemma 9
max{pj(F), pj,(Aj), j = 0,1,..., k — 1} = pJ(F) < a = pj(f).
By using Lemma 8, we obtain Aj(f) = Aj(f) = pj(f) and hence
Aj(f) = Aj(f) = pj(f) = a
with at most one exceptional solution f0 satisfying pi(f0) < a.
5. Conclusion
In this paper, by using the concepts of p-order and p-type, we have studied the growth of meromorphic solutions of higher order linear differential equations when among meromorphic coefficients having the maximal p-order, exactly one has its p-type stricly greater than others. Many previous results due to Chyzhykov-Semochko, Belaidi, Cao-Xu-Chen, Kinnunen have been extended. Now, it is interesting to study the growth of meromorphic solutions of such equations by using the concept of (a, ft)-order called the generalized order introduced by Sheremeta [20], see the recent paper of Mulyava-Sheremeta-Trukhan [17].
Acknowledgements
The authors are grateful to the referees for their many valuable remarks and suggestions which lead to the improvement of the original version of this paper.
This work was supported by the Directorate-General for Scientific Research and Technological Development (DGRSDT).
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