Electronic Journal of Differential Equations, Vol. 2010(2010), No. 164, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
GROWTH OF SOLUTIONS TO HIGHER ORDER LINEAR
HOMOGENEOUS DIFFERENTIAL EQUATIONS IN ANGULAR
DOMAINS
NAN WU
Abstract. In this article, we discuss the growth of meromorphic solutions
to higher order homogeneous differential equations in some angular domains,
instead of the whole complex plane.
1. Introduction and statement of main results
By a transcendental meromorphic function, we mean a function that is meromorphic on the whole complex plane, and is not a rational function; in other words, ∞
is an essential singular point. We assume the reader is familiar with the Nevanlinna
theory of meromorphic functions and basic notation such as: Nevanlinna characteristic T (r, f ), integrated counting function N (r, f ), and proximity function m(r, f ),
and the deficiency δ(a, f ) of f (z). For the details, see [4, 7]. The order λ and the
lower order µ are defined as follows:
log T (r, f )
log T (r, f )
λ(f ) = lim sup
, µ(f ) = lim inf
.
r→∞
log r
log r
r→∞
It is known the growth of meromorphic solutions of differential equations with
meromorphic coefficients in the complex plane C attracted a lot research. In this
article, we discuss the growth of meromorphic solutions of differential equations
with transcendental meromorphic coefficients in a proper subset of C. Let f (z) be
a meromorphic function in an angular region Ω(α, β) = {z : α ≤ arg z ≤ β}. Recall
the definition of Ahlfors-Shimizu characteristic in an angle (see [6]). Set
Ω(r) = Ω(α, β) ∩ {z : 1 < |z| < r} = {z : α < arg z < β, 1 < |z| < r}.
Define
S(r, Ω, f ) =
1
π
ZZ
Ω(r)
|f 0 (z)| 2
dσ,
1 + |f (z)|2
Z
T (r, Ω, f ) =
1
r
S(t, Ω, f )
dt.
t
The order and lower order of f on Ω are defined as follows
log T (r, Ω, f )
log T (r, Ω, f )
σα,β (f ) = lim sup
, µα,β (f ) = lim inf
.
r→∞
log r
log r
r→∞
2000 Mathematics Subject Classification. 30D10, 30D20, 30B10, 34M05.
Key words and phrases. Meromorphic solutions; Nevanlinna theory; order.
c
2010
Texas State University - San Marcos.
Submitted February 23, 2010. Published November 17, 2010.
Supported by grant 10871108 from the NSF of China.
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Remark 1.1. The order σα,β (f ) of a meormorphic function f on an angular region
here we give is reasonable, because T (r, C, f ) = T (r, f ) + O(1).
Nevanlinna theory on the angular domain plays an important role in value distribution of meromrorphic functions. Let us recall the following terms [3]:
Z
ω r 1
dt
tω Aα,β (r, f ) =
− 2ω {log+ |f (teiα )| + log+ |f (teiβ )|} ,
ω
π 1 t
r
t
Z β
2ω
log+ |f (reiθ )| sin ω(θ − α)dθ,
Bα,β (r, f ) = ω
πr α
X 1
|bn |ω Cα,β (r, f ) = 2
− 2ω sin ω(θn − α),
ω
|bn |
r
1<|bn |<r
where ω = π/(β − α), and bn = |bn |eiθn is a pole of f (z) in the angular domain
Ω(α, β), appears according to its multiplicity. The Nevanlinna’s angular characteristic is defined as follows:
Sα,β (r, f ) = Aα,β (r, f ) + Bα,β (r, f ) + Cα,β (r, f ).
Some articles define the order and lower order of f on Ω as:
σ α,β (f ) = lim sup
r→∞
log Sα,β (r, f )
,
log r
µα,β (f ) = lim inf
r→∞
log Sα,β (r, f )
.
log r
According to the inequality
Sα,β (r, f ) ≤ 2ω 2
T (r, Ω, f )
+ ω3
rω
Z
1
r
T (t, Ω, f )
dt + O(1),
tω+1
showed by Zheng [13, Theorem 2.4.7], if σα,β (r, f ) < ∞, then σ α,β (r, f ) < ∞.
We consider q pairs of real numbers {αj , βj } such that
− π ≤ α1 < β1 ≤ α2 < β2 ≤ . . . αq < βq ≤ π
(1.1)
∪qj=1 {z
and the angular domains X =
: αj ≤ arg z ≤ βj }. For a function f
meromorphic in the complex plane C, we define the order of f on X as
log T (r, X, f )
.
log r
r→∞
Pq
It is obvious that σαj ,βj (f ) ≤ σX (f ) ≤
j=1 σαj ,βj (f ). j = 1, 2, . . . , q. And
σX (f ) = +∞ if and only if there exists at least one 1 ≤ j0 ≤ q such that
σαj0 ,βj0 (f ) = +∞. We will establish the following results.
σX (f ) = lim sup
Theorem 1.2. Let A0 (z) be a meromorphic function in C with finite lower order
µ < ∞ and nonzero order 0 < λ ≤ ∞ and δ = δ(∞, A0 ) > 0. For q pair of real
numbers {αj , βj } satisfying (1.1) and
q
X
j=1
(αj+1 − βj ) <
p
4
arcsin δ/2
σ
(1.2)
where σ > 0 with µ ≤ σ ≤ λ. If Aj (z)(j = 1, 2, . . . , n) are meromorphic functions
in C with T (r, Aj ) = o(T (r, A0 )), then every solution f 6≡ 0 to the equation
An f (n) + An−1 f (n−1) + · · · + A0 f = 0
has the order σX (f ) = +∞ in X = ∪qj=1 {z : αj ≤ arg z ≤ βj }.
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If we remove the condition µ(A0 ) < ∞ in Theorem 1.2, we can establish the
following result.
Theorem 1.3. Let A0 (z) be a meromorphic function in C with nonzero order
0 < λ ≤ ∞ and δ(∞, A0 ) > 0. Suppose that for q directions arg z = αj (1 ≤ j ≤ q),
satisfying
−π ≤ α1 < α2 < · · · < αq < π, αq+1 = α1 + 2π,
Aj (z), j = 1, 2, . . . , n, are meromorphic functions in C with finite lower order and
T (r, Aj ) = o(T (r, A0 )). Then every solution f 6≡ 0 to the equation
An f (n) + An−1 f (n−1) + · · · + A0 f = 0
has order σX (f ) = +∞ in X = C\ ∪qj=1 {z : arg z = αj }.
The method in this paper was firstly used by Zheng [14] to investigate the growth
of transcendental meromorphic functions with radially distributed values.
2. Some auxiliary results
To prove the theorems, we give some lemmas. The following result is from
[12, 13, 14].
Lemma 2.1. Let f (z) be a transcendental meromorphic function with lower order
µ < ∞ and order 0 < λ ≤ ∞, then for any positive number µ ≤ σ ≤ λ and any set
E with finite measure, there exist a sequence {rn }, such that
/ E, limn→∞ rnn = ∞;
(1) rn ∈
T (rn ,f )
(2) lim inf n→∞ loglog
≥ σ;
rn
2t σ
(3) T (t, f ) < (1 + o(1))( rn ) T (rn /2, f ), t ∈ [rn /n, nrn ];
(4) T (t, f )/tσ−εn ≤ 2σ+1 T (rn , f )/rnσ−εn , 1 ≤ t ≤ nrn , εn = [log n]−2 .
We recall that {rn } is called the Pólya peaks of order σ outside E. Given a
positive function Λ(r) satisfying limr→∞ Λ(r) = 0. For r > 0 and a ∈ C, define
DΛ (r, a) = {θ ∈ [−π, π) : log+
1
> Λ(r)T (r, f )},
|f (reiθ ) − a|
and
DΛ (r, ∞) = {θ ∈ [−π, π) : log+ |f (reiθ )| > Λ(r)T (r, f )}.
The following result is called the spread relation, which was conjectured by Edrei
[2] and proved by Baernstein [1].
Lemma 2.2. Let f (z) be transcendental and meromorphic in C with the finite
lower order µ < ∞ and the positive order 0 < λ ≤ ∞ and has one deficient
b = C ∪ {∞}. Then for any sequence of Pólya peaks {rn } of order
values a ∈ C
σ > 0, µ ≤ σ ≤ λ and any positive function Λ(r) → 0 as r → +∞, we have
p
4
lim inf meas DΛ (rn , a) ≥ min{2π, arcsin δ(a, f )/2}.
n→∞
σ
To make it clearly, we give the definition of R-set on the complex plane C.
Definition
2.3. Let B(zn , rn ) = {z : |z −zn | < rn } be an open disk on the complex
P∞
plane. If n=1 rn < ∞, ∪∞
n=1 B(zn , rn ) is called an R-set.
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Lemma 2.4 ([8]). Let f be a meromorphic function on the angular region Ω(α, β)
with finite order ρ, let Γ = {(n1 , m1 ), (n2 , m2 ), . . . , (nj , mj )} denote a finite set of
distinct pair of integers which satisfying ni > mi ≥ 0 for i = 1, 2, . . . , j, and let
ε > 0 and δ > 0 be given constants. Then there exists K > 0 depending only on
f, ε, δ such that
f (n) (z) n−m
,
(2.1)
(m) < K|z|(n−m)(kδ +2ρ+1+ε) (sin kδ (ϕ − α − δ))−2
f (z)
for all (n, m) ∈ Γ and all z = reiϕ ∈ Ω(α + δ, β − δ) except for a R-set, that is, a
π
countable union of discs whose radii have finite sum, where kδ = β−α−2δ
.
To prove Theorem 1.3, we need a result from Edrei [2].
Lemma 2.5. Let f (z) be a meromorphic function with δ = δ(∞, f ) > 0. Then
given ε > 0, we have
1
,r ∈
/ F,
meas E(r, f ) > ε
T (r, f )[log r]1+ε
where
δ
T (r, f )}
4
and F is a set of positive real numbers with finite logarithmic measure depending
on ε.
E(r, f ) = {θ ∈ [−π, π) : log+ |f (reiθ )| >
3. Proof of the Theorems
Proof of Theorem 1.2. We suppose that there exists a nontrival meromorphic solution f such that σαj ,βj (f ) < +∞, j = 1, 2, . . . , q. In view of Lemma 2.4, there
exists a constant M > 0 not depending on z such that
(j) f (z) M
f (z) < |z| , j = 1, 2, . . . , n .
for all z ∈ Ω(αj + ε, βj − ε), j = 1, 2, . . . , q, except for a R-set E. For E, we can
define a set F = {r > 0|∃z ∈ E, s.t.|z| = r} thus
meas F < ∞.
(I) λ(A0 ) > µ(A0 ). Then λ(A0 ) > σ ≥ µ(A0 ). By the inequality (1.2), we can
take a real number ε > 0 such that
q
X
p
4
arcsin δ/2,
(αj+1 − βj + 2ε) + 2ε <
(3.1)
σ + 2ε
j=1
where αq+1 = 2π + α1 , and
λ(A0 ) > σ + 2ε > µ(A0 ).
Applying Lemma 2.1 to A(z) gives the existence of the Pólya peak {rn } of order
/ F , and then from Lemma 2.2 for sufficiently large n
σ + 2ε of A(z) such that rn ∈
we have
p
4
meas D(rn , ∞) >
arcsin δ/2 − ε.
(3.2)
σ + 2ε
We can assume for all the n, above holds. Set
K := meas(D(rn , ∞) ∩ ∪qj=1 (αj + ε, βj − ε)).
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Then from (3.1) and (3.2) it follows that
K ≥ meas(D(rn , ∞)) − meas([0, 2π)\ ∪qj=1 (αj + ε, βj − ε))
= meas(D(rn , ∞)) − meas(∪qj=1 (βj − ε, αj+1 + ε))
= meas(D(rn , ∞)) −
q
X
(αj+1 − βj + 2ε) > ε > 0.
j=1
It is easy to see that there exists a j0 such that for infinitely many n, we have
K
(3.3)
meas(D(rn , ∞) ∩ (αj0 + ε, βj0 − ε)) > .
q
We can assume for all the n, (3.3) holds. We define a real function by
T (rn , Aj )
log rn
2
,
; j = 1, 2, . . . , n ,
Λ(r) = max
T (rn , A0 ) T (rn , A0 )
for rn ≤ r < rn+1 . Obviously limr→∞ Λ(r) = 0 and
0
meas DΛ
(rn ) = meas{θ : rn eiθ ∈ E} = 0.
Set
0
Dn = (αj0 + ε, βj0 − ε)\DΛ
(rn ),
En = D(rn , ∞) ∩ (αj0 + ε, βj0 − ε).
Thus from the definition of D(r, ∞) it follows that
Z βj0 −ε
Z
log+ |A0 (rn eiθ )|dθ ≥
log+ |A0 (rn eiθ )|dθ
αj0 +ε
En
≥ meas(En )Λ(rn )T (rn , A0 )
K
> Λ(rn )T (rn , A0 ).
q
(3.4)
Thus, we have
Z βj0 −ε
log+ |A0 (rn eiθ )|dθ
αj0 +ε
Z
≤
n βj0 −ε X
αj0 +ε j=1
=
Z
Z
+
0 (r )
DΛ
n
Z
≤
≤
(j)
f (rn eiθ ) +
iθ
+ log |Aj (rn e )| dθ
log f (rn eiθ ) n
βj0 −ε X
+
n X
Dn
j=1
(j)
f (rn eiθ ) +
iθ
log + log |Aj (rn e )| dθ
f (rn eiθ ) +
log+ |Aj (rn eiθ )|dθ + O(log rn )
αj0 +ε j=1
n
X
T (rn , Aj ) + O(log rn )
j=1
≤ Λ2 (rn )T (rn , A0 ).
Therefore,
K
Λ(rn ) < Λ2 (rn ).
q
This contradicts that Λ(r) → 0.
(3.5)
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(II) λ(A0 ) = µ(A0 ). Then λ(A0 ) = σ = µ(A0 ). By the same argument as in
(I) with all the σ + 2ε replaced by σ, we can derive a contradiction. The proof is
complete.
Proof of Theorem 1.3. Applying Lemma 2.1 to A0 (z) confirms the existence of a
/ E and
sequence {rn } of positive numbers such that rn ∈
meas E(rn , A0 ) >
1
,
T ε (rn , A0 )[log rn ]1+ε
(3.6)
where E(rn , A0 ) is defined as in Lemma 2.5. Set
εn =
1
1
.
ε
2q + 1 T (rn , A0 )[log rn ]1+ε
Then for (3.6) it follows that
meas(E(rn , A0 ) ∩ ∪qj=1 (αj + εn , αj+1 − εn ))
≥ meas E(rn , A0 ) − meas(∪qj=1 (αj + εn , αj+1 − εn ))
≥ (2q + 1)εn − 2qεn = εn > 0.
so that there exists a j such that for infinitely many n, we have
εn
meas En >
,
q
(3.7)
where En = E(rn , A0 ) ∩ (αj + εn , αj+1 − εn ). We can assume that (3.7) holds for
all the n. Thus
Z αj+1 −εn
Z
+
iθ
log |A0 (rn e )|dθ ≥
log+ |A0 (rn eiθ )|dθ
αj +εn
En
δ
≥ meas(En ) T (rn , A0 )
4
δεn
T (rn , A0 ).
≥
4q
On the other hand,
Z αj+1 −εn
log+ |A0 (rn eiθ )|dθ <
αj +εn
n
X
T (rn , Aj ) + O(log rn )
(3.8)
(3.9)
j=1
Combining (3.8) and (3.9) gives
εn T (rn , A0 ) ≤
n
4q X
T (rn , Aj ) + O(log rn ),
δ j=1
so that
T 1−ε (rn , A0 ) ≤
n
X
4q(2q + 1)
[log rn ]1+ε
T (rn , Aj ) + O(log2+ε rn ),
δ
j=1
we have µ(A0 ) ≤ max1≤j≤q (µ(Aj ))/(1 − ε). By the same method as in Theorem
1.2, we obtain a contradiction, which completes the proof.
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Nan Wu
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China
E-mail address: wunan07@gmail.com