ROCKY MOUNTAIN
JOURNAL OF MATHEMATICS
Volume 44, Number 5, 2014
ON A LOGARITHMIC HARDY-BLOCH TYPE SPACE
XIAOMING WU AND SHANLI YE
ABSTRACT. In this paper, given 0 < p < ∞, we define a
logarithmic Hardy-Bloch type space
{
BHp,L = f (z) ∈ H(D) : ||f ||p,L
}
e
Mp (|z|, f ′ ) < ∞ .
= sup (1 − |z|) log
1 − |z|
z∈D
Then we mainly study the relation between BHp,L and
three classical spaces: Hardy space, Dirichlet type space and
VMOA. We also obtain some estimates on the growth of
f ∈ BHp,L .
1. Introduction. Let D = {z : |z| < 1} be the open unit disk in the
complex plane C, and let H(D) denote the set of all analytic functions
on D. For 0 ≤ r < 1, f (z) ∈ H(D), we set
(
)1/p
∫ 2π
1
iθ p
Mp (r, f ) =
|f (re )| dθ
, 0 < p < ∞,
2π 0
M∞ (r, f ) = max |f (reiθ )|.
0≤θ≤2π
For 0 < p ≤ ∞, the Hardy space H p consists of those functions
f ∈ H(D), for which
∥f ∥Hp = sup Mp (r, f ) < ∞.
0≤r<1
p
The Dirichlet type space Dp−1
consists of those functions f ∈ H(D),
2010 AMS Mathematics subject classification. Primary 30H10, 30H30, 30H35.
Keywords and phrases. Hardy space, Bloch space, Dirichlet type space.
The research was supported by National Natural Science Foundation of China
(grant No. 11271124) and Natural Science Foundation of Fujian Province, China
(grant No. 2009J01004).
The second author is the corresponding author.
Received by the editors on January 4, 2010, and in revised form on June 18,
2012.
DOI:10.1216/RMJ-2014-44-5-1669
c
Copyright ⃝2014
Rocky Mountain Mathematics Consortium
1669
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XIAOMING WU AND SHANLI YE
for which
∫
(1 − |z|)p−1 |f ′ (z)|p dA(z) < ∞,
D
where dA(z) = 1/π dx dy denotes the norm Lebesgue area measure on
p
if and only if
D. Hence, f ∈ Dp−1
∫
∫ 1
(1)
(1−|z|)p−1 |f ′ (z)|p dA(z) = 2π
r(1−r)p−1 Mpp (r, f ′ ) dr < ∞.
D
0
p
The Dp−1
is closely related to H p . A classical result of Littlewood
and Paley [9] (see also [10]) asserts that
p
H p ⊂ Dp−1
,
2 ≤ p < ∞.
On the other hand, see, e.g., [11, 14], we have
p
Dp−1
⊂ H p,
0 < p ≤ 2.
These inclusions are strict if p ̸= 2.
Given 0 < p ≤ ∞ and 0 ≤ α < ∞, we write BHp,α and BHp,L for
the spaces of those f (z) ∈ H(D), such that
{
BHp,α = f (z) ∈ H(D) : ||f ||p,α
}
α
= sup (1 − |z|) Mp (|z|, f ′ ) < ∞ ,
z∈D
{
BHp,L = f (z) ∈ H(D) : ||f ||p,L
}
e
′
= sup (1 − |z|) log
Mp (|z|, f ) < ∞ .
1 − |z|
z∈D
It is easy to prove that BHp,α and BHp,L are complete under the norms
||f ||pα = ||f ||p,α + |f (0)|,
||f ||pL = ||f ||p,L + |f (0)|.
When p ≥ 1, the two spaces above are Banach spaces.
For the two spaces above, it is evident that BHp,L ⊇ BHq,L
for 0 < p < q ≤ ∞. With the terminology just introduced, we
have BH∞,α = Bα and BH∞,L = BL , where Bα and BL denote
the α-Bloch space and the logarithmic Bloch space, respectively, the
ON A LOGARITHMIC HARDY-BLOCH TYPE SPACE
1671
properties on these two spaces are abundant. More information about
Bα and BL can be found in [12, 15, 16, 17, 18, 20, 21].
When 0 ≤ α < 1, the following result is due to Hardy and
Littlewood, see [8].
Theorem A. Let 0 ≤ α < 1 and 1 ≤ p ≤ ∞. Then
{
BHp,α = Λp1−α = f ∈ H p : ωp (t, f ) = O(t1−α ),
}
as t → 0 ,
where
(
ωp (t, f ) = sup
0<|h|≤t
1
2π
∫
2π
0
(
)1/p
)
( ) p
f ei(θ+h) − f eiθ dθ
,
t > 0, if 1 ≤ p < ∞,
)
(
(
)
( )
ω∞ (t, f ) = sup
ess sup f ei(θ+h) − f eiθ ,
t > 0.
0<|h|≤t
Thus, BHp,α is a mean Lipschitz space. On this basis, Blasco [2] and
Girela and Máquez [5] extend the result.
When α = 1, the following result is known (see [7] and [6]).
Theorem B. Let f ∈ H(D), if 0 < p < ∞ and f ∈ BHp,1 . Then
((
Mp (r, f ) = O
1
log
1−r
)β )
,
where
(i) β = 1/p, for 0 < p < 2,
(ii) β = 1/2 for 2 ≤ p < ∞.
Our main goal in this paper is to show that, if f ∈ BHp,L , whose
rate of growth Mp (r, f ′ ) (1 < p < ∞) is between the ones∩ for the
p
functions in BHp,α (0 < α < 1) and BHp,1 , then f ∈ (H p Dp−1
),
∩
p
p
see Theorem 2.1. We also characterize that BHp,L ⊂ (H
Dp−1 ) for
p ∈ (1, ∞) is strict. However, the containment is not true for 0 < p ≤ 1.
We also note that BHp,L ̸⊂ V M OA for every 0 < p < ∞. As for the
growth of f ∈ BHp,L , we give a sharp estimate.
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XIAOMING WU AND SHANLI YE
Throughout this paper, C, which may change from one occurrence
to the next, denotes a positive and finite constant only dependent on p
and α.
2. Proof of main results.
Theorem 2.1. Suppose 1 < p < ∞. Then BHp,L ⊂ (H p
∩
p
Dp−1
).
To complete the proof, we need the following three lemmas.
Lemma 2.1. [6] If 2 < p < ∞, then there is a constant Cp depending
only on p such that
(
(∫ 1
)1/2 )
2
′
||f ||Hp ≤ Cp |f (0)| +
(1 − r)Mp (r, f ) dr
,
0
for all f ∈ H(D).
Lemma 2.2. [7] If 0 < p ≤ 2, then there is a constant Cp depending
only on p such that
(
)
∫
p
p
p−1 ′
p
(1 − |z|) |f (z)| dA(z) ,
||f ||H p ≤ Cp |f (0)| +
D
p
for every f ∈ Dp−1
.
Lemma 2.3. If 0 < α, β < ∞, x ∈ (0, e], then f (x) = xα (log e/x)β
increases on (0, e1−β/α ], decreases on [e1−β/α , e].
The proof of this lemma is easy; we omit the details here.
Proof of Theorem 2.1. Take f ∈ BHp,L and assume, without loss
of generality, that f (0) = 0. When 2 < p < ∞, for 0 < r < 1, set
fr (z) = f (rz). Applying Lemma 2.1 to fr , we obtain that
∫ 1
Mp2 (r, f ) ≤ C
r2 (1 − s)Mp2 (rs, f ′ ) ds
∫
0
1
≤C
0
1−s
ds.
(1 − rs)2 (log e/(1 − rs))2
ON A LOGARITHMIC HARDY-BLOCH TYPE SPACE
1673
Since rs < s, Lemma 2.3 implies
∫ 1
1
Mp2 (r, f ) ≤ C
ds < ∞.
(1
−
s)(log
e/(1 − s))2
0
When 1 < p ≤ 2, using Lemma 2.2 yields
∫
Mpp (r, f ) ≤ C
rp (1 − |w|)p−1 |f ′ (rw)|p dA(w)
∫
D
1
∫
0
≤C
1
≤C
0
(1 − s)p−1
ds
(1 − rs)p (log e/(1 − rs))p
1
ds.
(1 − s)(log e/(1 − s))p
p
Hence f ∈ H . The assertion that f ∈ Dp−1
can be easily obtained
from (1) finishes the proof.
p
This theorem is not true for 0 < p ≤ 1 and p = ∞. Indeed, when
p = ∞, the space BH∞,L = BL . We take f (z) = log log e/(1 − z).
Then
(
)
1
M∞ (r, f ′ ) = O
,
(1 − r) log e/(1 − r)
that is, f ∈ BL , but f ∈
/ H ∞ . In the case 0 < p ≤ 1, we only prove
the following result.
Theorem 2.2. Suppose 0 < p ≤ 1. Let
f (z) =
(1 −
z)1/p
1
,
log e/(1 − z)
z ∈ D.
Then f ∈ BHp,L , but f ∈
/ H p.
The following three lemmas are needed in the proof of Theorem 2.2.
Lemma 2.4. If a > 1 and 0 < r < 1, then there exist two constants
(
)
2
1
C1 =
1−
a−1
(1 + π)a−1
and
C2 =
4π a
,
a−1
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XIAOMING WU AND SHANLI YE
such that
C1 (1 − r)
∫
1−a
≤
2π
iθ
ρe − r−a dt ≤ C2 (1 − r)1−a ,
ρ=
0
1
(1 + r).
2
The proof is similar to [3, Lemma in Chapter 4.6], so we omit the
details.
Lemma 2.5. For 0 < α, β < ∞ and z ∈ D, let
f (z) =
(1 − |z|)α (log e/(1 − |z|))
β
β
(|1 − z|)α (log e/|1 − z|)
.
Then f (z) ≤ 1 in the case β ≤ α log e/2, and f (z) ≤ (eα−β (6β)β )/2α αβ
in the case β > α log e/2.
Proof. For β ≤ α log e/2, by Lemma 2.3, we have
(
)β
(
)β
e
e
(1 − |z|)α log
≤ (|1 − z|)α log
,
1 − |z|
|1 − z|
for all z ∈ D,
which implies f (z) ≤ 1.
When β > α log e/2, let z ∈ D1 = {z ∈ D, |1 − z| < e1−(β/α) }, we
deduce that |f (z)| ≤ 1. On the other hand, the condition z ∈ D\D1
implies that
f (z) ≤
eα−β (β/α)β
β
2α (log e/2)
≤
6β eα−β (β/α)β
eα−β (6β)β
≤
.
2α
2α αβ
This finishes the proof.
The following lemma may be found in [8].
Lemma 2.6. Suppose that 0 < p < ∞ and f ∈ H p . Then
∫ 1
p
M∞
(r, f ) dr ≤ π||f ||pH p .
0
ON A LOGARITHMIC HARDY-BLOCH TYPE SPACE
1675
Proof of Theorem 2.2. Take the function
gc (z) =
zc
c.
(1 − z) (log 1/(1 − z))
Exercise 3 [3, Chapter I] says that gc ∈ H 1 (c > 1). However, set
z = reiθ ,
∫ 1
∫ 1
rc
1
M∞
(r, gc ) dr ≥
c dr = ∞,
0
0 (1 − r) (log 1/(1 − r))
for every 0 ≤ c ≤ 1,
which, by Lemma 2.6, implies gc (z) ∈
/ H 1 for 0 ≤ c ≤ 1.
For r = |z| ≥ 1/2, there exists a constant C > 0, such that
| log 1/(1 − z)| ≥ C. Therefore,
1/(1 − z) log(e/(1 − z))p |f (z)|p
= |g1 (z)|
z/(1 − z) log 1/(1 − z) | log 1/(1 − z)|
≥2
|(log e/(1 − z))p |
| log 1/(1 − z)|| log e/(1 − z)|1−p
≥2
1 + | log 1/(1 − z)|
(
)1−p
2C
e
≥
log
.
1+C
2
It follows that f ∈
/ H p . On the other hand,
f ′ (z) =
(1 −
1/p
1
−
2.
1+(1/p)
log e/(1 − z) (1 − z)
(log e/(1 − z))
z)1+(1/p)
We have
(
)p
1
1
1
|f (z)| ≤
+
|1 − z|1+p (log |e/(1 − z)|)p p log |e/(1 − z)|
(
)p
1
1
1
≤
+
.
p log e/2 |1 − z|1+p (log |e/(1 − z)|)p
′
p
Using Lemmas 2.4 and 2.5, we obtain that
∫ 2π
1
dθ
iθ |1+p (log |e/(1 − reiθ )|)p
|1
−
re
0
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XIAOMING WU AND SHANLI YE
∫
2π
1
dθ
iθ |p/2 (log |e/(1 − reiθ )|)p |1 − reiθ |1+(p/2)
|1
−
re
0
∫ 2π
C
1
≤
dθ
p
p/2
iθ
(1 − r) (log e/(1 − r)) 0 |1 − re |1+(p/2)
C
=
p.
(1 − r)p (log e/(1 − r))
=
This shows that f (z) ∈ BHp,L , and the proof is complete.
∩ p
We also note that BHp,L ⊂ (H p Dp−1
) for every p ∈ (1, ∞) is
proper. Indeed, we have the following theorem.
Theorem 2.3. Given
∩ p pwith 1 < p < ∞, there exists a function f
)\BHp,L .
which belongs to (H p Dp−1
Proof. Let
f (z) =
Then
(1 −
z)1/p
(
1
√
log(2e2
p )/(1
− z)
)1/√p ,
z ∈ D.
√
2e2 p
1
log
∼ log
1 − |z|
1 − |z|
as |z| → 1− .
Take gc in the proof of Theorem 2.2 with c =
Mpp (r, f ) ∼ M1 (r, g√p ),
√
p. Then
r = |z| → 1− ,
which implies f ∈ H p for every 1 < p < ∞.
p
As for proving f ∈ Dp−1
(1 < p < ∞), it can be deduced by directly
calculating
∫
1
0
Thus, we find f ∈ H p
r(1 − r)p−1 Mpp (r, f ′ ) dr < ∞.
∩
p
.
Dp−1
On the other hand, using Lemma 2.4 yields
∫ 2π
1
|f ′ (reiθ )|p dθ
2π 0
∫ 2π
1
1
√
√
≥
2π 0 |1 − reiθ |1+p | log(2e2 p )/(1 − reiθ )| p
ON A LOGARITHMIC HARDY-BLOCH TYPE SPACE
1677
(
)p
1
1
√
−√
dθ
p
p| log(2e2 p )/(1 − reiθ )|
∫ 2π
δ
1
1
√
√
≥
dθ
2
p
p
p
2π(2p) (log(2e
|1 − reiθ |1+p
)/(1 − r))
0
1
δ
√
√ ,
≥
p
p
2π(2p) (1 − r) (log(2e2 p )/(1 − r)) p
×
where δ > 0 is a constant. Therefore,
(1 − r) log
δ 1/p
e
log e/1 − r
√
√
√
Mp (r, f ′ ) ≥
p
2
1−r
2p(2π) (log(2e p )/(1 − r))1/ p
−→ ∞,
r → 1− ,
which implies f ∈
/ BHp,L , and this concludes the proof.
Remark 2.1. If we take
1
√
,
(1 − z)1/p log(2e2 p )/(1 − z)
f (z) =
z ∈ D.
Carefully checking the proofs of Theorems 2.2 and 2.3, we have
Mp (r, f ′ ) ≈
1
.
(1 − r) log e/(1 − r)
In [17], the second author showed that βL ⊆ V M OA, the vanishing
mean oscillation of analytic functions in D. So we want to know
whether BHp,L ⊆ V M OA for p ∈ (0, ∞). However, the answer is
negative. For more information for V M OA space, see [1, 4, 19].
Theorem 2.4. Suppose 0 < p < ∞, then
∫ z
1
f (z) =
dt ∈ BHp,L ,
1+1/p
log e/(1 − t)
0 (1 − t)
but f (z) ∈
/ V M OA.
Proof. The proof for f ∈ BHp,L is similar to Theorem 2.2, we omit
the details here.
1678
XIAOMING WU AND SHANLI YE
It is well known that the space
∩(
V M OA ⊂ BM OA ⊂ B
∩
)
Hp .
0<p<∞
We have
f ′ (r) =
(1 −
and then
(1 − r)f ′ (r) =
1
,
log e/(1 − r)
r)1+1/p
1
,
(1 − r)1/p log e/(1 − r)
which tends to infinity when r tends to 1. Hence, f ∈
/ B, it follows
that f ∈
/ V M OA.
Our next objective is to estimate the growth of f ∈ BHp,L (0 < p <
∞). We begin with two lemmas.
Lemma 2.7. Given p with 1 ≤ p < ∞ and r with 0 < r < 1, then
(2)
∫ r
(
)
1
1
2/p
ds
≤
2p
1
+
pe
.
1+1/p log e/(1−s)
1/p log e/(1−r)
(1−s)
(1−r)
0
Proof. Let r∗ = 1 − e1−2p . Using Lemma 2.3, we easily obtain
∫ r
∫ r
1
1
ds
≤
ds
1+1/p log e/(1 − s)
1+1/2p
0 (1 − s)
0 (1 − s)
≤ 2pe(2p−1)/2p
if r ≤ r∗ . When r > r∗ , we have
∫ r
1
ds
1+1/p log e/(1 − s)
(1
−
s)
0
∫ r∗
∫ r
1
1
=
ds+
ds
1+1/p log e/(1−s)
1+1/p log e/(1−s)
(1−s)
(1−s)
0
r∗
∫ r
1
1
≤ 2pe(2p−1)/2p +
ds
(1 − r)1+1/2p log e/(1 − r) 0 (1 − s)1+1/2p
1
≤ 2pe(2p−1)/2p + 2p
.
1/p
(1 − r) log e/(1 − r)
Hence, (2) holds.
ON A LOGARITHMIC HARDY-BLOCH TYPE SPACE
1679
Lemma 2.8. [8] Any function f ∈ H p (0 < p < ∞) can be expressed
in the form f (z) = f1 (z) + f2 (z), where f1 and f2 are nonvanishing H p
functions such that ∥fn ∥H p ≤ 2∥f ∥H p , n = 1, 2.
Theorem 2.5. If f ∈ BHp,L , (0 < p < ∞), then
|f (z)| ≤
(1 −
C∥f ∥pL
,
log e/(1 − r)
r)1/p
r = |z|,
where

)
(
 4πp(p − 1)(p−1)/p · 1 + p · e1/2p ,
C=
8(1 + e1/2 ),
 2+(3/p)−3p
2
/(1 − p),
1 < p < ∞,
p = 1,
0 < p < 1.
Proof. For r with 0 < r < 1, we have
∫ r
f ′ (seiθ )eiθ ds + f (0).
f (reiθ ) =
0
We set ρ = (1 + s)/2, by Cauchy formula
∫
∫ 2π ′ i(t+θ) i(t−θ)
1
f ′ (ζ)
ρ
f (ρe
)e
f ′ (seiθ ) =
dζ
=
dt.
2πi |ζ|=ρ ζ − seiθ
2π 0
ρeit − s
Case I: 1 < p < ∞. Let q be the conjugate index of p : (1/p)+(1/q) =
1, then, by Hölder’s inequality, Lemmas 2.4 and 2.7, we give
∫ r ∫ 2π ′ i(t+θ) i(t−θ)
1
|f (ρe
)e
|
dt ds
|f (reiθ )| ≤ |f (0)| +
2π 0 0
|ρeit − s|
( )1/q (
)1/q
1
4π q
≤ |f (0)| +
·
2π
q−1
∫ r
1
ds
Mp (ρ, f ′ ) ·
(1 − s)1/p
0
(
)
≤ |f (0)| + 4πp(p − 1)(p−1)/p 1 + p · e1/2p
∥f ∥p,L
(1 − r)1/p log e/(1 − r)
(
)
≤ 4πp(p − 1)(p−1)/p 1 + p · e1/2p
·
||f ||pL
.
(1 − r)1/p log e/(1 − r)
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XIAOMING WU AND SHANLI YE
Case II: p = 1. Using Lemma 2.7 again yields
∫ r ∫ 2π ′ i(t+θ) i(t−θ)
1
|f (ρe
)e
|
iθ
|f (re )| ≤ |f (0)| +
dt ds
2π 0 0
|ρeit − s|
∫ r
1
≤ |f (0)| + 4
ds · ||f ||p,L
2 log e/(1 − s)
(1
−
s)
0
||f ||pL
≤ 8(1 + e1/2 )
.
(1 − r) log e/(1 − r)
Case III: 0 < p < 1. If f ′ (z) ̸= 0 in z ∈ D, then the function
F (z) = (f ′ (z))p is analytic, and f ∈ BHp,L gives
M1 (r, F ) = {Mp (r, f ′ )} ≤
||f ||pp,L
p
(1 − r)p (log e/(1 − r))
p.
By the Cauchy formula, we find
∫ 2π
2||f ||pp,L
F (ρeit )eit
ρ
dt|
≤
|F (reiθ )| = |
p,
2π 0
ρeit − z
(1 − r)1+p (log e/(1 − r))
which implies that
|f ′ (reiθ )| ≤
Then
M1 (r, f ′ ) =
1
2π
√
p
2||f ||p,L
.
(1 − r)1+1/p log e/(1 − r)
∫
2π
|f ′ (reiθ )|1−p |f ′ (reiθ )|p dθ
0
≤ {M∞ (r, f ′ )}
1−p
≤
(1 −
We deduce
{Mp (r, f ′ )}
p
2(1−p)/p
· ||f ||pp,L .
log e/(1 − r)
r)1/p
∫
∫
|f ′ (ρei(t+θ) )ei(t−θ) |
dt ds
|ρeit − s|
0
0
p
||f ||p,L
≤ |f (0)| + 22/p
·
1/p
1 − p (1 − r) log e/(1 − r)
|f (reiθ )| ≤ |f (0)| +
≤
1
2π
r
2π
22/p
||f ||pL
·
.
1 − p (1 − r)1/p log e/(1 − r)
ON A LOGARITHMIC HARDY-BLOCH TYPE SPACE
1681
If f ′ (z) has zeros, we fix R < 1 and use Lemma 2.8 to write
f ′ (Rz) = f1′ (z) + f2′ (z),
where f1 and f2 do not vanish and
∥fn′ ∥H p ≤ 2Mp (R, f ′ ) ≤
2||f ||p,L
,
(1 − R) log e/(1 − R)
n = 1, 2.
Since fn′ (z) ̸= 0 (n = 1, 2), it follows that
|f ′ (R2 eiθ )| ≤ |f1′ (Reiθ )| + |f2′ (Reiθ )|
≤
Then
|f ′ (reiθ )| ≤
22+1/p ||f ||p,L
.
(1 − R)1+1/p log e/(1 − R)
23+2/p ||f ||p,L
,
(1 − r)1+1/p log e/(1 − r)
which implies that
|f (reiθ )| ≤
22+3/p−3p
||f ||pL
·
.
1−p
(1 − r)1/p log e/(1 − r)
This completes the proof of Theorem 2.5.
For 0 < p < ∞, we set
{
(
)
}
1
∞
Hp,L
= f ∈ H(D), |f (z)| = sup
<
∞,
|z|
=
r
.
(1−r)1/p log e/(1−r)
∞
Then BHp,L and the classic Bloch space B1 are included in Hp,L
. It
∞
∞
p
turns out that neither Hp,L nor Hp,L is contained in H for every
1 < p < ∞. Indeed,
f (z) =
∞
∑
n
z2 ,
z∈D
n=0
∞
is in B1 , which implies that f ∈ Hp,L
, but not in H p . On the other
hand, taking the function f (z) in Theorem 2.4, we find that f ∈ H p
∞
but not f ∈
/ Hp,L
.
We set U here to be the class of all univalent functions in D. Then
Prawitz, see [13, page 17], deduced the following theorem.
1682
XIAOMING WU AND SHANLI YE
∫1 p
Theorem D. Suppose that 0 < p < ∞. If f ∈ U and 0 M∞
(r, f ) dr <
∞, then f ∈ H p . Using this result, we can give the following theorem.
Theorem 2.6. Suppose that 1 < p < ∞ and f ∈ U
f ∈ H p.
∩
∞
Hp,L
. Then
Acknowledgments. We thank the referee for numerous stylistic
corrections.
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Department of Mathematics, Fujian Normal University, Fuzhou 350007,
P.R. China and Putian No. 1 Middle School of Fujian Province, Putian
351100, P.R. China
Email address: [email protected]
Department of Mathematics, Fujian Normal University, Fuzhou 350007,
P.R. China
Email address: [email protected]