Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 73, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OSCILLATION OF MEROMORPHIC SOLUTIONS TO LINEAR
DIFFERENTIAL EQUATIONS WITH COEFFICIENTS OF
[p, q]-ORDER
HONG-YAN XU, JIN TU
Abstract. We study the relationship between “small functions” and the derivative of solutions to the higher order linear differential equation
f (k) + Ak−1 f (k−1) + · · · + A0 f = 0,
(k ≥ 2)
Here Aj (z) (j = 0, 1, . . . , k − 1) are entire functions or meromorphic functions
of [p, q]-order.
1. Introduction and statement of main results
The study of oscillation theory for linear differential equations in the complex
plane C was started by Bank and Laine [2, 3]. After their well-known work, many
important results have been obtained, see for example [19, 20].
We assume that the reader knows the standard notations and the fundamental
results of the Nevanlinna value distribution theory of meromorphic functions [12,
15]. In addition, we use σ(f ), λ(f ) and λ(f ) to denote the order, the exponent of
convergence of the zero-sequence, and the exponent of convergence of the nonzero
zero sequence of a meromorphic function f (z), respectively. We also denote by τ (f )
the type of an entire
R function f (z) with
R 0 < σ(f ) = σ < +∞ (see [15]).
We use mE = E dt and ml E = E dt
t to denote the linear measure and the
logarithmic measure of a set E ⊂ [1, +∞), respectively. We denote by S(r, f ) any
quantity satisfying S(r, f ) = o(T (r, f )), as r → +∞, possibly outside of a set with
finite linear measure. A meromorphic function ψ(z) is called a small function with
respect to f if T (r, ψ) = S(r, f ).
For results on the growth of solutions to equations of the form
f 00 + A(z)f 0 + B(z)f = 0,
(1.1)
with A(z) and B(z)(6≡ 0) are entire functions, the reader is referred to [1, 7, 8, 11,
14].
In 1996, Kwon [18] investigated the hyper-order of the solutions of (1.1) and
obtained the following result.
2000 Mathematics Subject Classification. 34M10, 30D35.
Key words and phrases. Linear differential equation; oscillation; small function; [p, q]-order.
c
2014
Texas State University - San Marcos.
Submitted June 20, 2013. Published March 16, 2014.
1
2
H.-Y. XU, J. TU
EJDE-2014/73
Theorem 1.1 ([18]). Let A(z) and B(z) be entire functions such that σ(A) < σ(B)
or σ(B) < σ(A) < 1/2. Then every solution f 6≡ 0 of (1.1) satisfies σ2 (f ) ≥
max{σ(A), σ(B)}.
In 2006, Chen and Shon [9] investigated the zeros concerning small functions and
fixed points of solutions of second order linear differential equations and obtained
the following results.
Theorem 1.2 ([9]). Let Aj (z) 6≡ 0 (j = 1, 2) be entire functions with σ(Aj ) < 1,
suppose that a, b are complex numbers that satisfy ab 6= 0 and arg a 6= arg b or
a = cb (0 < c < 1). If ϕ(z) 6≡ 0 is an entire function of finite order, then every
non-trivial solution f of equation
f 00 + A1 (z)eaz f 0 + A2 (z)ebz f = 0
satisfies λ(f − ϕ) = λ(f 0 − ϕ) = λ(f 00 − ϕ) = ∞.
Theorem 1.3. [9] Let A1 (z) 6≡ 0, ϕ(z) 6≡ 0, Q(z) be entire functions with σ(A1 ) <
1, 1 < σ(Q) < ∞ and σ(ϕ) < ∞, then every non-trivial solution f of equation
f 00 + A1 (z)eaz f 0 + Q(z)f = 0
satisfies λ(f − ϕ) = λ(f 0 − ϕ) = λ(f 00 − ϕ) = ∞, where a 6= 0 is a complex number.
In 2012, Wu and Chen [24] investigate the problem on the fixed-points of solutions of some second order differential equation with transcendental entire function
coefficients and obtained the following theorems.
Theorem 1.4 ([24, Theorem 1]). Let Aj (z) 6≡ 0(j = 0, 1) be entire functions,
P (z) be a polynomial satisfying σ(A1 ) < deg P (z) and 0 < σ(A0 ) < 1/2, and let
ϕ(z)(6≡ 0) be an entire function of finite order. Then every non-trivial solution f
of equation
f 00 + A1 (z)eP (z) f 0 + A0 (z)f = 0
satisfies λ(f − ϕ) = ∞.
Theorem 1.5 ([24, Theorem 2]). Under the assumptions of Theorem 1.4, every
non-trivial solution f of the equation
f 00 + A1 (z)eP (z) f 0 + A0 (z)f = 0
satisfies
(i) λ(f − z) = λ(f 0 − z) = λ(f 00 − z) = σ(f ) = ∞;
(ii) g(z) has infinitely many fixed points and λ(g − z) = ∞, where g(z) =
d0 f (z) + d1 f 0 (z) + d2 f 00 (z), d0 d2 6= 0.
An interesting subject arises naturally about the problems of the zeros concerning
small function and fixed points of solutions of differential equations
f (k) + Ak−1 f (k−1) + · · · + A0 f = 0,
(k ≥ 2)
(1.2)
where Aj (z) (j = 0, 1, . . . , k − 1) are entire functions.
In 2000s, Belaı̈di [4], Belaı̈di and El Farissi [6] (see also [10, 22, 23]) investigated
the fixed points and the relationship between small functions and differential polynomials of solutions of (1.2) and obtained some results which improve Theorem
1.3.
EJDE-2014/73
OSCILLATION OF MEROMORPHIC SOLUTIONS
3
Recently, the growth of solutions of higher order linear differential equation with
meromorphic coefficients of [p, q]-order was studied and some results were obtained
in [5, 21].
In this article, we study the zeros of small functions and the fixed points of
solutions to equation (1.2) with entire or meromorphic coefficients of [p, q]-order
and obtain some results that extend the work of Chen and Belaı̈di.
Before stating our theorems, we introduce the concepts of entire functions of
[p, q]-order (see [16, 17, 21]). Juneja and co-authors [16, 17] introduced the concept of entire functions of [p, q]-order, and studied some of their properties for p, q
integers satisfying p > q ≥ 1.
Definition 1.6. If f (z) is a transcendental entire function, the [p, q]-order of f (z)
is defined by
σ[p,q] (f ) = lim sup
r→∞
logp T (r, f )
logp+1 M (r, f )
= lim sup
,
logq r
logq r
r→∞
where p, q are two integers and p > q ≥ 1.
Remark 1.7. For sufficiently large r ∈ [1, ∞), we define logi+1 r = logi (log r)
(i ∈ N) and expi+1 r = exp(expi r) (i ∈ N) and exp0 r = r = log0 r, exp−1 r = log r.
Definition 1.8. The [p, q]-type of an entire function f of [p, q]-order σ (0 < σ < ∞)
is defined by
logp M (r, f )
.
τ[p,q] = τ[p,q] (f ) = lim sup
(logq−1 r)σ
r→∞
And the [p, q] exponent of convergence of the zero sequence of f is defined by
λ[p,q] = λ[p,q] (f ) = lim sup
r→∞
logp n(r, f1 )
logq r
= lim sup
r→∞
logp N (r, f1 )
logq r
,
and the [p, q] exponent of convergence of the distinct zero sequence of f is defined
by
logp n(r, f1 )
logp N (r, f1 )
λ[p,q] = λ[p,q] (f ) = lim sup
= lim sup
.
logq r
logq r
r→∞
r→∞
Let ϕ(z) be an entire function with σ[p,q] (ϕ) < σ[p,q] (f ), the [p, q] exponent of
convergence of zeros and distinct zeros of f (z) − ϕ(z) are defined to be
λ[p,q] (f − ϕ) = lim sup
r→∞
1
logp N (r, f −ϕ
)
logq r
, λ[p,q] (f − ϕ) = lim sup
r→∞
1
logp N (r, f −ϕ
)
logq r
,
especially if ϕ(z) = z, we use λ[p,q] (f − z) and λ[p,q] (f − z) to denote the [p, q] exponent of convergence of fixed points and distinct fixed points of f (z), respectively.
Next we state our main results.
Theorem 1.9. It Aj (z) (j = 0, 1, . . . , k − 1) are entire functions and satisfy one
of the following two conditions:
(i) max{σ[p,q] (Aj ) : j = 1, 2, . . . , k − 1} < σ[p,q] (A0 ) < ∞;
(ii) max{σ[p,q] (Aj ) : j = 1, 2, . . . , k − 1} ≤ σ[p,q] (A0 ) < ∞ and
max{τ[p,q] (Aj )|σ[p,q] (Aj ) = σ[p,q] (A0 ) > 0} = τ1 < τ[p,q] (A0 ) = τ ,
4
H.-Y. XU, J. TU
EJDE-2014/73
then for every solution f ≡
6 0 of (1.2) and for any entire function ϕ(z) 6≡ 0 satisfying
σ[p+1,q] (ϕ) < σ[p,q] (A0 ). Moreover
λ[p+1,q] (f − ϕ) = λ[p+1,q] (f 0 − ϕ) = λ[p+1,q] (f 00 − ϕ)
= λ[p+1,q] (f (i) − ϕ) = σ[p+1,q] (f )
= σ(A0 ),
(i ∈ N).
Throughout this paper we assume that A0 does not vanish identically.
Theorem 1.10. If Aj (z), j = 0, 1, . . . , k − 1 are meromorphic functions satisfying
max{σ[p,q] (Aj ) : j = 1, 2, . . . , k − 1} < σ[p,q] (A0 ) and δ(∞, A0 ) > 0, then for every
meromorphic solution f 6≡ 0 of (1.2) and for any meromorphic function ϕ(z) 6≡ 0
satisfying σ[p+1,q] (ϕ) < σ[p,q] (A0 ), we have
λ[p+1,q] (f (i) − ϕ) = λ[p+1,q] (f (i) − ϕ) ≥ σ[p,q] (A0 )(i = 0, 1, . . . ),
where f (0) = f .
Example 1.11. For the equation
−e2z
e2z + ez − 1 0
f +
f = 0,
(1.3)
z
1−e
1 − ez
z
we can easily see that this equation has a solution f (z) = ee + ez . The func2z
z
2z
2z
z
+e −1 −e
+e −1
−e2z
tions e 1−e
, 1−ez are meromorphic and satisfy σ( e 1−e
) = σ( 1−e
z
z
z ) = 1 and
1
−e2z
−e2z
z
δ ∞, 1−ez = 2 . Taking ϕ(z) = e , then σ[2,1] (ϕ) < σ[1,1] ( 1−ez ). Thus, we get that
f 00 +
2z
z
−e
λ[2,1] (f 0 − ϕ) = λ[2,1] (ee ez ) = 0 6= 1 = σ[1,1] ( 1−e
z ).
For p > q ≥ 1, we have the following example.
Example 1.12. Consider the equation
f 00 + A1 f 0 + A0 f = 0,
where
z
z
z
z
1 + ez − 2ee + e2e − 2ee ez + e3e ez
,
A1 = −
(1 − eez )2
z
z
e3e e2z − e2e e2z
A0 =
.
(1 − eez )2
Obviously, A0 , A1 are meromorphic functions, σ[2,1] (A1 ) = σ[2,1] (A0 ) = 1 and
ez
δ(∞, A0 ) > 0. By calculating, the equation (1.12) has a solution f (z) = ee +
z
z
ee . Taking ϕ(z) = ee ez , then σ[3,1] (ϕ) < σ[2,1] (A0 ). Thus, we can get that
ez
z
λ[3,1] (f 0 − ϕ) = λ[3,1] (ee ee ez ) = 0 6= 1 = σ[2,1] (A0 ).
From Theorems 1.9 and 1.10, we obtain the following corollaries.
Corollary 1.13. Under the assumptions of Theorem 1.9, if ϕ(z) = z, for every
solution f 6≡ 0 of (1.2), we have
λ[p+1,q] (f − z) = λ[p+1,q] (f 0 − z) = λ[p+1,q] (f 00 − z)
= λ[p+1,q] (f (i) − z) = σ[p+1,q] (f )
= σ[p,q] (A0 ),
(i ∈ N).
Corollary 1.14. Under the assumptions of Theorem 1.10, if ϕ(z) = z, for every
meromorphic solution f 6≡ 0 of (1.2), we have λ[p+1,q] (f (i) −z) = λ[p+1,q] (f (i) −z) ≥
σ[p+1,q] (A0 ), (i = 0, 1, . . . ), where f (0) = f .
EJDE-2014/73
OSCILLATION OF MEROMORPHIC SOLUTIONS
5
2. Preliminary results
To prove our theorems, we require the following lemmas.
Lemma 2.1 ([25, Lemma 2.1]). Assume f 6≡ 0 is a solution of (1.2), set g = f −ϕ,
then g satisfies the equation
g (k) + Ak−1 g (k−1) + · · · + A0 g = −[ϕ(k) + Ak−1 ϕ(k−1) + · · · + A0 ϕ].
(2.1)
Lemma 2.2 ([25, Lemma 2.2]). Assume f 6≡ 0 is a solution of equation (1.2), set
g1 = f 0 − ϕ , then g1 satisfies the equation
(k)
(k−1)
1
g1 + Uk−1
g1
1
+ · · · + U01 g1 = −[ϕ(k) + Uk−1
ϕ(k−1) + · · · + U01 ϕ],
where Uj1 = A0j+1 + Aj −
A00
A0 Aj+1 ,
(2.2)
j = 0, 1, 2, . . . , k − 1 and Ak ≡ 1.
Lemma 2.3 ([25, Lemma 2.5]). Assume f 6≡ 0 is a solution of equation (1.2), set
gi = f (i) − ϕ, then gi satisfies the equation
(k)
gi
(k−1)
i
+ Uk−1
gi
0
i
+ · · · + U0i gi = −[ϕ(k) + Uk−1
ϕ(k−1) + · · · + U0i ϕ],
i−1
where Uji = Uj+1
+ Uji−1 −
0
U0i−1
i−1
U0
(2.3)
i−1
Uj+1
, j = 0, 1, 2, . . . , k − 1, Uki−1 ≡ 1 and i ∈ N.
Lemma 2.4 ([21, Lemma 3.9]). Let f (z) be an entire function of [p, q]-order, then
σ[p,q] (f ) = σ[p,q] (f 0 ).
Lemma 2.5 ([21, Lemma 3.10]). Let f (z) be an entire function of [p, q]-order satisfying σ[p,q] (f ) = σ2 , then there exists a set E ⊂ [1, +∞) with infinite logarithmic
measure such that for all r ∈ E, we have
logp T (r, f )
lim
= σ2 , r ∈ E.
r→∞
logq r
Lemma 2.6. Let A0 (z), A1 (z), . . . , Ak−1 (z) be entire functions with [p, q]-order and
satisfy max{σ[p,q] (Aj ) : j = 1, 2, . . . , k − 1} = σ1 < σ[p,q] (A0 ) < ∞, and set
Uj1 = A0j+1 + Aj −
A00
Aj+1
A0
and
0
U0i−1 i−1
Uj+1 ,
U0i−1
where j = 0, 1, 2, . . . , k − 1, Ak ≡ 1, Uki−1 ≡ 1 and i ∈ N. Then there exists a set E
with infinite logarithmic measure such that for r ∈ E, we have
0
i−1
Uji = Uj+1
+ Uji−1 −
logp m(r, U0i )
r→∞
logq r
σ[p,q] (A0 ) = lim
max1≤j≤k−1 {logp m(r, Uji )}
> lim sup
= σ1 .
logq r
r→∞
(2.4)
Proof. We will use the inductive method to prove it.
A0
First, when i = 1, it follows that Uj1 = A0j+1 +Aj − A00 Aj+1 for j = 0, 1, 2, . . . , k−1
and Ak ≡ 1. When j = 0, that is, U01 = A01 + A0 −
m(r, U01 ) ≤ m(r, A1 ) + m(r, A0 ) + m r,
A01 A1
A00
A 0 A1 .
+ m r,
Then, we have
A00 + O(1).
A0
(2.5)
6
H.-Y. XU, J. TU
From A0 = −A01 + U01 +
A00
A 0 A1 ,
EJDE-2014/73
we have
m(r, A0 ) ≤ m(r, A1 ) + m(r, U01 ) + m r,
A01 A0 + m r, 0 + O(1).
A1
A0
(2.6)
When j 6= 0, from the definitions of Uj1 (j = 1 . . . , k), we have
A0 m r, Uj1 ≤ m(r, Aj+1 ) + m(r, Aj ) + m r, 0
A0
0
Aj+1 + m r,
+ O(1), j = 1, 2, . . . , k − 1.
Aj+1
(2.7)
Since A0 (z), . . . , Ak−1 (z) are entire functions with max{σ[p,q] (Aj ) : j = 1, 2, . . . , k −
1} < σ[p,q] (A0 ) < ∞ and (2.7), we have
max {m(r, Uj1 )t}
1≤j≤k−1
≤
max {m(r, Aj ) + o(m(r, A0 ) + O(log(rT (r, f )))} + O(1),
(2.8)
1≤j≤k−1
holds for all r ∈ E1 − E2 (where E1 is a set of infinite logarithmic measure and
E2 is a set of finite linear measure). From (2.5), (2.6), (2.8) and Lemma 2.5, there
exists a set E ⊂ [1, +∞) with infinite logarithmic measure such that
logp m(r, U01 )
r→∞
logq r
σ[p,q] (A0 ) = lim
> σ1 = lim sup
r→∞
≥ lim sup
r→∞
max1≤j≤k−1 {logp m(r, Aj )}
logq r
(2.9)
max1≤j≤k−1 {logp m(r, Uj1 )}
, r ∈ E.
logq r
Now, suppose that (2.4) holds for i ≤ n(n ∈ N), thus, there exists a set E with
infinite logarithmic measure such that
logp m(r, U0n )
r→∞
logq r
σ[p,q] (A0 ) = lim
> lim sup
r→∞
max1≤j≤k−1 {logp m(r, Ujn )}
= σ1 .
logq r
(2.10)
Next, we prove that (2.4) holds for i = n + 1. From the assumptions of this lemma,
U n0 n
n 0
we have Ujn+1 = Uj+1
+ Ujn − U0n Uj+1
, (j = 0, 1, 2, . . . , k − 1) and Ukn ≡ 1 for
0
i = n + 1. Thus, when j = 0, it follows that U0n+1 = U1n 0 + U0n −
have
U0n 0 n
U0n U1 .
U n0
U n0
m r, U0n+1 ≤ m (r, U0n ) + m(r, U1n ) + m(r, 0n ) + m(r, 1n ) + O(1).
U0
U1
And since U0n = −U1n 0 + U0n+1 +
U0n 0 n
U0n U1 ,
Then, we
(2.11)
we have
m(r, U0n ) ≤ m(r, U0n+1 ) + m(r, U1n ) + m(r,
U0n 0
U n0
) + m(r, 1n ) + O(1).
n
U0
U1
(2.12)
EJDE-2014/73
OSCILLATION OF MEROMORPHIC SOLUTIONS
7
When j 6= 0, it follows from the definitions of Ujn+1 (j = 1, 2, . . . , k − 1) and Ukn ≡ 1
that
n 0
Uj+1
U n0
n
m(r, Ujn+1 ) ≤ m(r, Uj+1
) + m(r, Ujn ) + m(r, n ) + m(r, 0n ) + O(1). (2.13)
Uj+1
U0
From (2.10)–(2.13), there exists a set E with infinite logarithmic measure such that
logp m r, U0n+1
logp m (r, U0n )
lim
= lim
= σ[p,q] (A0 )
r→∞
r→∞
logq r
logq r
max1≤j≤k−1 {logp m r, Ujn }
(2.14)
> σ1 = lim sup
logq r
r→∞
= lim sup
r→∞
max1≤j≤k−1 {logp m(r, Ujn+1 )}
,
logq r
r ∈ E.
Thus, the proof is complete.
Lemma 2.7. Let Hj (z) (j = 0, 1, . . . , k − 1) be meromorphic functions of finite
[p, q]-order. If
max1≤j≤k−1 {logp m(r, Hj )}
lim sup
= β1
logq r
r→∞
and there exists a set E1 with infinite logarithmic measure such that
logp m(r, H0 )
= β2 > β1
r→∞
logq r
lim
holds for all r ∈ E1 , then every meromorphic solution f 6≡ 0 of
f (k) + Hk−1 f (k−1) + · · · + H1 f 0 + H0 f = 0
(2.15)
satisfies σ[p+1,q] (f ) ≥ β2 .
Proof. Assume that f (z) 6≡ 0 is a meromorphic solution of (2.15). From (2.15), we
have
m(r, H0 ) ≤ m(r,
k−1
X
f (k)
f0
) + · · · + m(r, ) +
m(r, Hj ) + O(1).
f
f
j=1
(2.16)
By the logarithmic derivative lemma and (2.16), we have
m(r, H0 ) ≤ O{log rT (r, f )} +
k−1
X
m(r, Hj ),
r 6∈ E2 ,
(2.17)
j=1
where E2 ⊂ [1, +∞) is a set with finite linear measure. From the assumptions of
Lemma 2.7, there exists a set E1 with infinite logarithmic measure such that for all
|z| = r ∈ E1 − E2 , we have
expp {(β2 − ε) logq r} ≤ O{log rT (r, f )} + (k − 1) expp {(β1 + ε) logq r},
where 0 < 2ε < β2 − β1 . From (20), we have σ[p+1,q] (f ) ≥ β2 .
(2.18)
Lemma 2.8 ([13]). Let f (z) be a transcendental meromorphic function and α > 1
be a given constant. Then for any given ε > 0, there exists a set E7 ⊂ [1, ∞)
that has finite logarithmic measure and a constant M > 0 that depends only on α
8
H.-Y. XU, J. TU
EJDE-2014/73
and (m, n)(m, n ∈ {0, . . . , k} with m < n) such that for all z satisfying |z| = r 6∈
[0, 1] ∪ E7 , we have
n−m
f (n) (z) ≤ M T (αr, f ) (logα r) log T (αr, f )
.
r
f (m) (z)
Lemma 2.9 ([21, Lemma 3.13]). Let f (z) be an entire function of [p, q]-order
satisfying σ[p,q] (f ) = σ, τ[p,q] (f ) = τ , 0 < σ < ∞, 0 < τ < ∞, then for any given
β < τ , there exists a set E4 ⊂ [1, +∞) that has infinite logarithmic measure such
that for all r ∈ E4 , we have
logp M (r, f ) > β(logq−1 r)σ .
Lemma 2.10. Let A0 (z), A1 (z), . . . , Ak−1 (z) be entire functions with finite [p, q]order and satisfy max{σ[p,q] (Aj ) : j = 1, 2, . . . , k − 1} ≤ σ[p,q] (A0 ) = σ2 < ∞ and
max{τ[p,q] (Aj ) |σ[p,q] (Aj ) = σ[p,q] (A0 ) > 0} = τ1 < τ[p,q] (A0 ) = τ , and let Uj1 , Uji
be as stated in Lemma 2.6. Then for any given ε(0 < 2ε < τ − τ1 ), there exists a
set E5 with infinite logarithmic measure such that
|Uji | ≤ expp {(τ1 + ε)(logq−1 r)σ2 },
|U0i | ≥ expp {(τ − ε)(logq−1 r)σ2 },
(2.19)
where i ∈ N and j = 1, 2, . . . , k − 1.
Proof. We will use the induction method for this proof.
(i) First, we prove that Uji (j = 0, 1, . . . , k − 1) satisfy (2.19) when i = 1. From
the definition Uj1 = A0j+1 + Aj −
have
A00
A0 Aj+1 (j
6= 0) and U01 = A01 + A0 −
0
A
A0
|U01 | ≥ −|A1 | | 1 | + | 0 | + |A0 |
A1
A0
A00
A 0 A1 ,
we
(2.20)
and
|Uj1 | ≤ |Aj+1 | |
A0j+1
A0 | + | 0 | + |Aj |,
Aj+1
A0
j = 1, 2, . . . , k − 1; Ak ≡ 1.
(2.21)
From Lemma 2.8, Lemma 2.9 and (2.20)–(2.21), for any ε(0 < 4ε < τ − τ1 ), there
exists a set E5 with infinite logarithmic measure such that
ε
|U01 | ≥ −2M expp {(τ1 + )(logq−1 r)σ2 }(T (2r, A0 ))2
8
ε
+ expp {(τ − )(logq−1 r)σ2 }
4
2
ε
ε
≥ −2M expp {(τ1 + )(logq−1 r)σ2 } expp {(σ2 + )(logq 2r)}
8
8
ε
σ2
(logq−1 r) }
+ expp { τ −
4
ε
≥ expp { τ −
(logq−1 r)σ2 }
2
(2.22)
EJDE-2014/73
OSCILLATION OF MEROMORPHIC SOLUTIONS
9
and
ε
|Uj1 | ≤ 2M expp {(τ1 + )(logq−1 r)σ2 }(T (2r, A0 ))2
4
ε
+ expp {(τ1 + )(logq−1 r)σ2 }
4
2
ε
ε
≤ 2M expp {(τ1 + )(logq−1 r)σ2 } expp {(σ2 + )(logq 2r)}
(2.23)
4
8
ε
+ expp {(τ1 + )(logq−1 r)σ2 }
4
ε
≤ expp {(τ1 + )(logq−1 r)σ2 }, j 6= 0,
2
where M > 0 is a constant, not necessarily the same at each occurrence.
(ii) Next, we show that Uji (j = 0, 1, 2, . . . , k − 1) satisfy (2.19) when i = 2. From
0
U02 = U11 + U01 −
Uk1 ≡ 1, we have
and
0
U01
U01
0
1
U11 and Uj2 = Uj+1
+ Uj1 −
0
U01
U01
1
Uj+1
(j = 0, 1, . . . , k − 1) and
0
U 10
U1 |U02 | ≥ |U01 | − |U11 | | 11 | + | 01 |
U1
U0
0
U1 0
U1 j+1
1
|Uj2 | ≤ |Uj1 | + |Uj+1
| | 1 | + | 01 | ,
Uj+1
U0
(2.24)
j = 1, 2, . . . , k − 1.
(2.25)
By the conclusions in (i), Lemma 2.8 and Lemma 2.9, (2.22)–(2.23), for all |z| =
r ∈ E5 , we have
2
ε
ε
|U02 | ≥ −2M expp (τ1 + )(logq−1 r)σ2 expp (σ2 + )(logq 2r)
2
8
ε
σ2
(2.26)
+ expp { τ −
(logq−1 r) }
2
≥ expp {(τ − ε)(logq−1 r)σ2 }
and
ε
ε
|Uj2 | ≤ 2M expp {(τ1 + )(logq−1 r)σ2 } expp {(σ2 + )(logq 2r)}
2
8
ε
σ2
+ expp {(τ1 + )(logq−1 r) }
2
≤ expp {(τ1 + ε)(logq−1 r)σ2 }, j 6= 0.
(2.27)
(iii) Now, suppose that (2.19) holds for i ≤ n(n ∈ N). Thus, for any given
ε(0 < 4ε < τ − τ1 ), there exists a set E5 with infinite logarithmic measure such
that
|Uji | ≤ expp {(τ1 + ε)(logq−1 r)σ2 }, |U0i | ≥ expp {(τ − ε)(logq−1 r)σ2 },
where i ≤ n and j = 1, 2, . . . , k − 1. From U0n+1 = U1n 0 + U0n −
n 0
Ujn+1 = Uj+1
+ Ujn −
U0n 0
U0n
U0n 0
U0n
(2.28)
U1n and
n
Uj+1
(j = 0, 1, . . . , k − 1) and Ukn ≡ 1, we have
U n0
U n0 |U0n+1 | ≥ |U0n | − |U1n | | 1n | + | 0n |
U1
U0
(2.29)
and
Un 0
U n0 j+1
n
|Ujn+1 | ≤ |Ujn | + |Uj+1
| | n | + | 0n | ,
Uj+1
U0
j = 1, 2, . . . , k − 1.
(2.30)
10
H.-Y. XU, J. TU
EJDE-2014/73
Then, from Lemma 2.8, Lemma 2.9 and (2.28)–(2.30), for all |z| = r ∈ E5 , we have
2
ε
|Ujn+1 | ≤ 2M expp {(τ1 + ε)(logq−1 r)σ2 } expp {(σ2 + )(logq 2r)}
8
(2.31)
+ expp {(τ1 + ε)(logq−1 r)σ2 }
≤ expp {(τ1 + 2ε)(logq−1 r)σ2 },
j 6= 0,
and
ε
|U0n+1 | ≥ −2M exp{(τ1 + ε)(logq−1 r)σ2 }(expp {(σ2 + )(logq 2r)})2
8
+ expp {(τ − ε)(logq−1 r)σ2 }
(2.32)
σ2
≥ expp {(τ − 2ε)(logq−1 r) }.
Thus, the proof is complete.
Lemma 2.11. Let Bj (z) (j = 0, 1, . . . , k − 1) be meromorphic functions such that
max{σ[p,q] (Bj ) : j = 1, 2, . . . , k − 1} = σ4 < σ[p,q] (B0 ) = σ3 and δ := δ(∞, B0 ) =
0)
limr→∞ m(r,B
T (r,B0 ) > 0. Then every meromorphic solution f 6≡ 0 of equation
f (k) + Bk−1 f (k−1) + · · · + B1 f 0 + B0 f = 0
(2.33)
satisfies σ[p+1,q] (f ) ≥ σ3 .
Proof. Let f 6≡ 0 be a meromorphic solution of equation (2.33). Then from (2.33),
we have
k−1
X
f (k)
f (k−1)
f0
m(r, Bj ) + O(1)
m(r, B0 ) ≤ m(r,
) + m(r,
) + · · · + m(r, ) +
f
f
f
j=1
≤ O{log rT (r, f )} +
k−1
X
T (r, Bj ),
r 6∈ E6 ,
j=1
(2.34)
where E6 ⊂ [1, +∞) is a set with finite linear measure. By Lemma 2.5, there exists
a set E with infinite logarithmic measure such that for all |z| = r ∈ E, we have
logp T (r, B0 )
= σ3 .
r→∞
logq r
lim
(2.35)
Since δ := δ(∞, B0 ) > 0, then for any given ε(0 < 2ε < min{δ, σ3 − σ4 }) and for all
r ∈ E, by (37), we have
m(r, B0 ) ≥ (δ − ε) expp {(σ3 − ε) logq r}.
(2.36)
From (2.34) and (2.36), we have
(δ −ε) expp {(σ3 −ε) logq r} ≤ O{log rT (r, f )}+(k −1) expp {(σ4 +ε) logq r}, (2.37)
where r ∈ E − E6 . From (2.37), we obtain σ[p+1,q] (f ) ≥ σ3 = σ[p,q] (B0 ).
Lemma 2.12. Let Bj (z), j = 0, 1, . . . , k − 1 be meromorphic functions of finite
[p, q] order. If there exist positive constants σ5 , β3 , β4 (0 < β3 < β4 ) and a set E8
with infinite logarithmic measure such that
max{|Bj (z)| : j = 1, 2, . . . , k − 1} ≤ expp {β3 (logq−1 r)σ5 },
and
|B0 (z)| ≥ expp {β4 (logq−1 r)σ5 }
EJDE-2014/73
OSCILLATION OF MEROMORPHIC SOLUTIONS
11
hold for all |z| = r ∈ E8 , then every meromorphic solution f 6≡ 0 of (2.33) satisfies
σ[p+1,q] (f ) ≥ σ5 .
Proof. Suppose that f 6≡ 0 is a meromorphic function of (2.33). Then it follows
that
k−1
X
f (k)
f (j)
|B0 (z)| ≤ |
|+
|.
(2.38)
|Bj (z)||
f
f
j=1
By Lemma 2.8, there exists a set E7 with finite logarithmic measure such that for
all |z| = r 6∈ E7 , we have
|
f (j)
| ≤ M [T (2r, f )]2j ,
f
j = 1, 2, . . . , k.
(2.39)
By (2.38), (2.39) and the assumptions of Lemma 2.12, for all |z| = r ∈ E8 − E7 , we
have
expp {β4 (logq−1 r)σ5 } ≤ M k[T (2r, f )]2k expp {β3 (logq−1 r)σ5 }.
Since 0 < β3 < β4 and by (2.40), we have σ[p+1,q] (f ) ≥ σ5 .
(2.40)
Lemma 2.13 ([21, Lemma 3.12]). Let A0 , A1 , . . . , Ak−1 , F 6≡ 0 be meromorphic
functions, if f is a meromorphic solution of the equation
f (k) + Ak−1 f (k−1) + · · · + A0 f = F,
satisfying max{σ[p,q] (F ), σ[p,q] (Aj ); j = 0, 1, . . . , k − 1} < σ[p,q] (f ), then we have
σ[p,q] (f ) = λ[p,q] (f ) = λ[p,q] (f ).
Lemma 2.14 ([21, Theorem 2.3]). Let Aj (z) (j = 0, 1, . . . , k−1) be entire functions
satisfying max{σ[p,q] (Aj ) : j = 1, 2, . . . , k − 1} ≤ σ[p,q] (A0 ) < ∞ and
max{τ[p,q] (Aj )|σ[p,q] (Aj ) = σ[p,q] (A0 ) > 0} < τ[p,q] (A0 ).
Then every nontrivial solution f of (1.2) satisfies σ[p+1,q] (f ) = σ[p,q] (A0 ).
3. Proofs of Theorems
Proof of Theorem 1.9. We will consider two cases as follows.
Case 1. Suppose that max{σ[p,q] (Aj ) : j = 1, 2, . . . , k − 1} < σ[p,q] (A0 ) < ∞.
(i) First, we prove that λ[p+1,q] (f −ϕ) = σ[p+1,q] (f ). Assume that f is a nontrivial
solution of (1.2), from [21, Theorem 2.2], we have σ[p+1,q] (f ) = σ[p,q] (A0 ). Set
g = f − ϕ. Since σ[p+1,q] (ϕ) < σ[p,q] (A0 ), then σ[p+1,q] (g) = σ[p+1,q] (f ) = σ[p,q] (A0 )
and λ[p+1,q] (g) = λ[p+1,q] (f −ϕ). By Lemma 2.1, we get that g satisfies the equation
(2.1). Set F = ϕ(k) + Ak−1 ϕ(k−1) + · · · + A0 ϕ. If F ≡ 0, then from [21], we have
σ[p+1,q] (ϕ) = σ[p,q] (A0 ), a contradiction. Then F 6≡ 0. From Lemma 2.4 and
assumption of Case 1, we have
σ[p+1,q] (F ) ≤ max{σ[p+1,q] (ϕ), σ[p+1,q] (A0 )} = max{σ[p+1,q] (ϕ), 0}.
Since σ[p+1,q] (ϕ) < σ[p,q] (A0 ), we have
max{σ[p+1,q] (F ), σ[p+1,q] (Aj ) : j = 0, 1, 2, . . . , k − 1} < σ[p+1,q] (f ).
By Lemma 2.13, we have λ[p+1,q] (g) = λ[p+1,q] (g) = σ[p+1,q] (g) = σ[p,q] (A0 ). Thus,
we have
λ[p+1,q] (f − ϕ) = λ[p+1,q] (f − ϕ) = σ[p+1,q] (f ) = σ[p,q] (A0 ).
12
H.-Y. XU, J. TU
EJDE-2014/73
(ii) Second, we prove that λ[p+1,q] (f 0 − ϕ) = σ[p+1,q] (f ). Set g1 = f 0 − ϕ,
then σ[p+1,q] (g1 ) = σ[p+1,q] (f ) = σ[p,q] (A0 ). From Lemma 2.2, we get that g1
1
satisfies the equation (2.2). Set F1 = ϕ(k) + Uk−1
ϕ(k−1) + · · · + U01 ϕ, where Uj1 (j =
0, 1, . . . , k − 1) are stated as in Lemma 2.2. If F1 ≡ 0, from Lemma 2.6 and
Lemma 2.7, we have σ[p+1,q] (ϕ) ≥ σ[p,q] (A0 ), a contradiction with σ[p+1,q] (ϕ) <
σ[p,q] (A0 ). Hence F1 6≡ 0. From the definition of Uj1 (j = 0, 1, . . . , k − 1), we
have σ[p+1,q] (Uj1 ) ≤ σ[p+1,q] (Aj ) j = 0, 1, . . . , k − 1. Thus, we can get σ[p+1,q] (F1 ) ≤
max{σ[p+1,q] (ϕ), σ[p+1,q] (Uj1 ) : j = 0, 1, . . . , k−1}. Since σ[p+1,q] (ϕ) < σ[p,q] (A0 ), we
have max{σ[p+1,q] (F1 ), σ[p+1,q] (Uj1 ) : j = 0, 1, . . . , k − 1} < σ[p,q] (A0 ) = σ[p+1,q] (g1 ).
By Lemma 2.13, we obtain
λ[p+1,q] (f 0 − ϕ) = λ[p+1,q] (f 0 − ϕ) = σ[p+1,q] (f ).
(iii) We will prove that λ[p+1,q] (f (i) − ϕ) = σ[p+1,q] (f ), (i > 1, i ∈ N). Set
gi = f (i) − ϕ, then σ[p+1,q] (gi ) = σ[p+1,q] (f ) = σ[p,q] (A0 ). From Lemma 2.3, we
i
have gi satisfies equation (2.3). Set Fi = ϕ(k) + Uk−1
ϕ(k−1) + · · · + U0i ϕ, where
i
Uj (j = 0, 1, . . . , k −1; i ∈ N) are stated as in Lemma 2.3. If Fi ≡ 0, from Lemma 2.6
and Lemma 2.7, we have σ[p+1,q] (ϕ) ≥ σ[p,q] (A0 ), a contradiction with σ[p+1,q] (ϕ) <
σ[p,q] (A0 ). Hence Fi 6≡ 0. By using the same argument as in Case 1(ii), we can get
λ[p+1,q] (f (i) − ϕ) = λ[p+1,q] (f (i) − ϕ) = σ[p+1,q] (f ).
Case 2. Suppose that max{σ[p,q (Aj ) : j = 1, 2, . . . , k − 1} ≤ σ[p,q] (A0 ) < ∞ and
max{τ[p,q] (Aj )|σ[p,q] (Aj ) = σ[p,q] (A0 ) > 0} < τ[p,q] (A0 ).
(i) We first prove that λ[p+1,q] (f −ϕ) = σ[p+1,q] (f ). Since f is a nontrivial solution
of (1.2), by Lemma 2.14, we have σ[p+1,q] (f ) = σ[p,q] (A0 ) > 0. Set g = f − ϕ.
Since ϕ 6≡ 0 is an entire function satisfying σ[p+1,q] (ϕ) < σ[p,q] (A0 ), then we have
σ[p+1,q] (g) = σ[p+1,q] (f ) = σ[p,q] (A0 ) and λ[p+1,q] (g) = λ[p+1,q] (f −ϕ). From Lemma
2.1, we get that g satisfies equation (2.1). We will affirm F 6≡ 0. If F ≡ 0, by
Lemma 2.14, we get σ[p+1,q] (ϕ) = σ[p,q] (A0 ), a contradiction. Hence F 6≡ 0. From
the assumptions of Case 2, we get
max{σ[p+1,q] (F ), σ[p+1,q] (Aj ) : j = 0, 1, . . . , k − 1} < σ[p+1,q] (g) = σ[p,q] (A0 ).
From Lemma 2.13, we have
λ[p+1,q] (f − ϕ) = λ[p+1,q] (f − ϕ) = σ[p+1,q] (f ) = σ[p,q] (A0 ).
(ii) Now we prove that λ[p+1,q] (f 0 − ϕ) = σ[p+1,q] (f ). Let g1 = f 0 − ϕ. Since
σ[p+1,q] (ϕ) < σ[p,q] (A0 ), we have σ[p+1,q] (g1 ) = σ[p+1,q] (f ) = σ[p,q] (A0 ). By Lemma
2.2, we get that g1 satisfies equation (2.2). If F1 ≡ 0, from Lemma 2.10 and
Lemma 2.12, we have σ[p+1,q] (ϕ) ≥ σ[p,q] (A0 ). Then we can get a contradiction
with σ[p+1,q] (ϕ) < σ[p,q] (A0 ). Therefore, we have F1 6≡ 0. By (2.2) and Lemma
2.13, we have
λ[p+1,q] (f 0 − ϕ) = λ[p+1,q] (f 0 − ϕ) = σ[p+1,q] (f ) = σ[p,q] (A0 ).
Similar to the arguments as in Case 1 (iii) and by using Lemmas 2.3, 2.10 and 2.12,
we obtain
λ[p+1,q] (f (i) − ϕ) = λ[p+1,q] (f (i) − ϕ) = σ[p+1,q] (f ) = σ[p,q] (A0 ),
Thus, the proof is complete.
(i ∈ N).
EJDE-2014/73
OSCILLATION OF MEROMORPHIC SOLUTIONS
13
Proof of Theorem 1.10. According to the conditions of Theorem 1.2, we can easily
obtain the conclusions by using the similar argument as in Theorem 1.9 and Lemma
2.11.
Acknowledgments. This work was supported by the NSF of China (11301233,
61202313) and the Natural Science Foundation of Jiang-Xi Province in China
(20132BAB211001 and 20132BAB211002).
References
[1] I. Amemmiya, M. Ozawa; Non-existence of finite order solutions of ω 00 + e−z ω 0 + Q(z)ω = 0,
Hokkaido Math. J. 10 (1981), 1-17.
[2] S. Bank, I. Laine; On the oscillation theory of f 00 + Af = 0 where A is entire, Trans. Amer.
Math. Soc. 273 (1982), 351-363.
[3] S. Bank, I. Laine; On the zeros of meromorphic solutions of second order linear differential
equations, Comment. Math. Helv. 58(1983), 656-677.
[4] B. Belaı̈di; Growth and oscillation theory of solutions of some linear differential equations,
Matematički Vesnik 60 (4) (2008), 233-246.
[5] B. Belaı̈di; Growth of solutions to linear differential equations with analytic coefficients of
[p, q]-order in the unit disc, Electron. J. Diff. Equ. 2011 (2011), No. 156, 1-11.
[6] B. Belaı̈di, A. El Farissi; Differential polynomials generated by some complex linear differential equations with meromorphic coefficients, Glasnik Matematički 43 (2) (2008), 363-373.
[7] Z. X. Chen; On the hyper order of solutions of some second order linear differential equations,
Acta Math. Sinica B 18(1) (2002), 79-88.
[8] Z. X. Chen; The growth of solutions of the differential equation f 00 + e−z f 0 + Q(z)f = 0, Sci.
China Ser. A 31(2001), 775-784 (in Chinese).
[9] Z. X. Chen, K. H. Shon; The relation between solutions of a class of second order differential equation with functions of small growth, Chinese. Ann. Math. 27(A4) (2006), 431442(Chinese).
[10] A. EL Farissi , M. Benbachir; Oscillation of fixed points of solutions of some linear differential
equations, Electron. J. Diff. Equ. 2013 (2013), No. 41, pp. 1-9.
[11] M. Frei; Uber die subnormalen losungen der differentialgeichung ω 00 + e−z ω 0 + (konst.)ω = 0,
Comment Math. Helv. 36 (1962), 1-8.
[12] S. A. Gao, Z. X. Chen, T. W. Chen; Oscillation Theory on Linear Differential Equations,
Wuhan: Press in Central China Institute of Technology, 1998.
[13] G. G. Gundersen; Estimates for the logarithmic derivative of a meromorphic function, plus
similar estimates, J. London Math. Soc. 305(2) (1988), 88-104.
[14] G. G. Gundersen; On the question of whether f 00 + e−z f 0 + B(z)f = 0 can admit a solution
f 6≡ 0 of finite order, Proc R.S.E. 102 A (1986), 9-17.
[15] W. K. Hayman; Meromorphic Functions, Clarendon, Oxford, 1964.
[16] O.P. Juneja, G. P. Kapoor, S. K. Bajpai; On the (p, q)-order and lower (p, q)-order of an
entire function, J. Reine Angew. Math. 282 (1976), 53-67.
[17] O. P. Juneja, G. P. Kapoor, S. K. Bajpai; On the (p, q)-type and lower (p, q)-type of an entire
function, J. Reine Angew. Math. 290 (1977), 180-190.
[18] K. H. Kwon; On the growth of entire functions satisfying second order linear differential
equation, Bull. Korean. Math. Soc. 3 (1996), 487-496.
[19] I. Laine; Nevanlinna theory and complex differential equations, W. de Gruyter, Berlin, 1993.
[20] I. Laine; Complex differential equations, Handbook of Differential Equations: Ordinary Differential Equations, 4 (2008), 269-363.
[21] J. Liu, J. Tu, L. Z. Shi; Linear differential equations with entire coefficients of [p, q]-order in
the complex plane, J. Math. Anal. Appl. 372(2010), 55-67.
[22] M. S. Liu, X. M. Zhang; Fixed points of meromorphic solutions of higher order linear differential equations, Ann.Acad Sci. Fenn. Ser. A. I. Math. 31 (2006), 191-211.
[23] J. Tu, T. Long; Oscillation of complex high order linear differential equations with coefficients
of finite iterated order, Electon. J. Qualitative Theory of Diff. Equ. 66 (2009), 1-13.
[24] Z. J. Wu, Y. X. Chen; The fixed points of solutions of some second order differential equation,
Acta Math. Scientia 32 (A) 2012, 779-784.
14
H.-Y. XU, J. TU
EJDE-2014/73
[25] H. Y. Xu, J. Tu, X. M. Zheng; On the hyper exponent of convergence of zeros of f (j) − ϕ of
higher order linear differential equations, Advances in Difference Equations 2012, 2012: 114,
doi: 10.1186/1687-1847-2012-114.
Hong-Yan Xu
Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen,
Jiangxi 333403, China
E-mail address: xhyhhh@126.com
Jin Tu
Institute of Mathematics and informatics, Jiangxi Normal University, Nanchang, Jiangxi
330022, China
E-mail address: tujin2008@sina.com