OPTIMAL BOUNDS FOR A TOADER-TYPE MEAN IN TERMS
OF ONE-PARAMETER QUADRATIC AND CONTRAHARMONIC
MEANS
WEI-MAO QIAN, YU-MING CHU, AND YING-QING SONG
Abstract. In this paper, we present the best possible parameters λ1 , µ1 , λ2 , µ2 ∈
(1/2, 1) such that the double inequalities
Q(a, b; λ1 ) < T [A(a, b), Q(a, b)] < Q(a, b; µ1 ),
C(a, b; λ2 ) < T [A(a, b), Q(a, b)] < C(a, b; µ2 )
p
R
hold for all a, b > 0 with a 6= b, where T (a, b) = 2 0π/2 a2 cos2 θ + b2 sin2 θdθ/π,
p
A(a, b) = (a + b)/2, Q(a, b) =
(a2 + b2 )/2, C(a, b) = (a2 + b2 )/(a + b),
Q(a, b; p) = Q[pa + (1 − p)b, pb + (1 − p)a] and C(a, b; q) = C[qa + (1 − q)b, qb +
(1 − q)a] are respectively the Toader, arithmetic, quadratic, contraharmonic,
one-parameter quadratic and one-parameter contraharmonic means of a and
b.
1. Introduction
Let M (a, b) be a one-parameter symmetric bivariate mean, p ∈ [0, 1], q ∈ R
and a, b > 0 with a 6= b. Then the one-parameter mean M (a, b; p), pth generalized
Seiffert mean Sp (a, b), qth Gini mean Gq (a, b), qth power mean Mq (a, b), qth Lehmer
mean Lq (a, b), harmonic mean H(a, b), geometric mean G(a, b), arithmetic mean
A(a, b), quadratic mean Q(a, b), contraharmonic mean C(a, b), Toader mean T (a, b)
[1], centroidal mean C(a, b) are respectively defined by
M (a, b; p) = M [pa + (1 − p)b, pb + (1 − p)a],
(
p(a−b)
0 < p ≤ 1,
2p(a−b) ,
arctan[ a+b ]
Sp (a, b) =
a+b
,
p = 0,
 2
1/(p−2)
q−1
q−1
 a +b
, q 6= 2,
a+b
Gq (a, b) =
 a b 1/(a+b)
a b
,
q = 2,
(
q
q 1/q
a +b
, q 6= 0,
√ 2
Mq (a, b) =
ab,
q = 0,
(1.1)
2010 Mathematics Subject Classification. 26E60.
Key words and phrases. Toader mean, arithmetic mean, quadratic mean, contraharmonic
mean.
This research was supported by the Natural Science Foundation of China under Grants
11371125, 11401191 and 61374086, the Natural Science Foundation of Zhejiang Province under
Grant LY13A010004, the Natural Science Foundation of Hunan Province under Grant 12C0577
and the Natural Science Foundation of the Zhejiang Broadcast and TV University under Grant
XKT-15G17.
1
Optimal Bounds for a Toader-Type Mean in Terms of One-Parameter Quadratic and
Contraharmonic Means
2
√
2ab
H(a, b) =
, G(a, b) = ab,
a+b
r
2
a+b
a + b2
a 2 + b2
A(a, b) =
, Q(a, b) =
, C(a, b) =
,
(1.2)
2
2
a+b
Z
2 π/2 p 2 2
2(a2 + ab + b2 )
T (a, b) =
a cos θ + b2 sin2 θdθ, C(a, b) =
.
π 0
3(a + b)
It is well known that Sp (a, b), Gq (a, b), Mq (a, b) and Lq (a, b) are continuous and
strictly increasing with respect to p ∈ [0, 1] and q ∈ R for fixed a, b > 0 with a 6= b,
and the inequalities
Lq (a, b) =
aq+1 + bq+1
,
aq + bq
H(a, b) = M−1 (a, b) = L−1 (a, b) < G(a, b) = M0 (a, b) = L−1/2 (a, b)
(1.3)
< A(a, b) = M1 (a, b) = L0 (a, b) < T (a, b) < C(a, b)
< Q(a, b) = M2 (a, b) < C(a, b) = L1 (a, b)
hold for all a, b > 0 with a 6= b.
The Toader mean T (a, b) is well known in mathematical literature for many
years, it satisfies
T (a, b) = RE a2 , b2 ,
where
Z
1 ∞ [a(t + b) + b(t + a)]t
RE (a, b) =
dt
π 0 (t + a)3/2 (t + b)3/2
stands for the symmetric complete elliptic integral of the second kind (see [2-4]),
therefore it can’t be expressed in terms of the elementary transcendental functions.
π/2
π/2
R
R
Let r ∈ (0, 1), K(r) =
(1 − r2 sin2 θ)−1/2 dθ and E(r) =
(1 − r2 sin2 θ)1/2 dθ
0
0
be respectively the complete elliptic integrals of the first and second kind. Then
K(0+ ) = E(0+ ) = π/2, K(1− ) = +∞, E(1− ) = 1, K(r) and E(r) satisfy the
derivatives formulas (see [5, Appendix E, p. 474-475])
E(r) − (1 − r2 )K(r) dE(r)
E(r) − K(r) d(K(r) − E(r))
rE(r)
dK(r)
=
,
=
,
=
,
2
dr
r(1 − r )
dr
r
dr
1 − r2
√
√
the values K( 2/2) and E( 2/2) can be expressed as (see [6, Theorem 1.7])
√ !
√ !
Γ2 14
4Γ2 34 + Γ2 41
2
2
√
√
= 1.854 · · · , E
= 1.350 · · ·
K
=
=
2
2
4 π
8 π
R∞
where Γ(x) = 0 tx−1 e−t dt is the Euler gamma function, and the Toader mean
T (a, b) can be rewritten as

q
 2a
b 2

1− a
, a > b,
 πE
q
T (a, b) =
(1.4)

a 2

E
1
−
,
a
<
b.
 2b
π
b
Recently, the Toader mean T (a, b) has attracted the attention of many researchers. Vuorinen [7] conjectured that the inequality
T (a, b) > M3/2 (a, b)
holds for all a, b > 0 with a 6= b. This conjecture was proved by Qiu and Shen [8],
and Barnard, Pearce and Richards [9], respectively.
3
Wei-Mao Qian, Yu-Ming Chu and Ying-Qing Song
Alzer and Qiu [10] presented a best possible upper power mean bound for the
Toader mean as follows:
T (a, b) < Mlog 2/(log π−log 2) (a, b)
for all a, b > 0 with a 6= b.
Neuman [2], and Kazi and Neuman [3] proved that the inequalities
√
√
4(a + b) ab + (a − b)2
(a + b) ab − ab
< T (a, b) <
,
AGM (a, b)
8AGM (a, b)
q
q
√
√
√
√
1
2
2
2
2
T (a, b) <
(2 + 2)a + (2 − 2)b + (2 + 2)b + (2 − 2)a
4
hold for all a, b > 0 with a 6= b, where AGM (a, b) is the arithmetic-geometric mean
of a and b.
In [11-13], the authors presented the best possible parameters λ1 , µ1 ∈ [0, 1] and
λ2 , µ2 , λ3 , µ3 ∈ R such that the double inequalities Sλ1 (a, b) < T (a, b) < Sµ1 (a, b),
Gλ2 (a, b) < T (a, b) < Gµ2 (a, b) and Lλ3 (a, b) < T (a, b) < Lµ3 (a, b) hold for all
a, b > 0 with a 6= b.
Let λ, µ, α, β ∈ (1/2, 1). Then Chu, Wang and Ma [14], and Hua and Qi [15]
proved that the double inequalities
C(a, b; λ) < T (a, b) < C(a, b; µ),
C(a, b; α) < T (a, b) < C(a, b; β)
p
hold for all a, b > 0 with a 6= b ifpand only if λ ≤ 3/4, µ ≥ 1/2 + π(4 − π)/(2π),
√
α ≤ 1/2 + 3/4 and β ≥ 1/2 + 12/π − 3/2.
In [16-20], the authors proved that the double inequalities
α1 Q(a, b) + (1 − α1 )A(a, b) < T (a, b) < β1 Q(a, b) + (1 − β1 )A(a, b),
Qα2 (a, b)A(1−α2 ) (a, b) < T (a, b) < Qβ2 (a, b)A(1−β2 ) (a, b),
α3 C(a, b) + (1 − α3 )A(a, b) < T (a, b) < β3 C(a, b) + (1 − β3 )A(a, b),
α4
1 − α4
1
β4
1 − β4
+
<
<
+
,
A(a, b) C(a, b)
T (a, b)
A(a, b) C(a, b)
α5 C(a, b) + (1 − α5 )H(a, b) < T (a, b) < β5 C(a, b) + (1 − β5 )H(a, b),
α6 [C(a, b) − H(a, b)] + A(a, b) < T (a, b) < β6 [C(a, b) − H(a, b)] + A(a, b),
α7 C(a, b) + (1 − α7 )A(a, b) < T (a, b) < β7 C(a, b) + (1 − β7 )A(a, b),
1
α8
1 − α8
β8
1 − β8
+
<
<
+
,
A(a, b) C(a, b)
T (a, b)
A(a, b) C(a, b)
α9 Q(a, b) + (1 − α9 )H(a, b) < T (a, b) < β9 Q(a, b) + (1 − β9 )H(a, b),
α10
1 − α10
1
β10
1 − β10
+
<
<
+
H(a, b)
Q(a, b)
T (a, b)
H(a, b)
Q(a, b)
√
hold for all a, b > 0 with a 6= b if and only if α1 ≤ 1/2, β1 ≥ (4 − π)/[( 2 − 1)π],
α2 ≤ 1/2, β2 ≥ 4 − 2 log π/ log 2, α3 ≤ 1/4, β3 ≥ 4/π − 1, α4 ≤ π/2 − 1, β4 ≥ 3/4,
α5 ≤ 5/8, β5 ≥ 2/π, α6 ≤ 1/8,
√ β6 ≥ 2/π −1/2, α7 ≤ 3/4, β7 ≥ 12/π −3, α8 ≤ π −3,
β8 ≥ 1/4, α9 ≤ 5/6, β9 ≥ 2 2/π, α10 ≤ 0 and β10 ≥ 1/6.
Very recently, Song et. al. [21] proved that the double inequality
Mp (a, b) < T [A(a, b), Q(a, b)] < Mq (a, b)
holds for
√ all a, b > 0 with a 6= b if and only if p ≤ 2 log 2/[2 log π − log 2 −
2 log E( 2/2)] = 1.3930 · · · and q ≥ 3/2.
Optimal Bounds for a Toader-Type Mean in Terms of One-Parameter Quadratic and
Contraharmonic Means
4
From (1.1) and (1.2) we clearly see that both the functions x → Q(a, b; x) and
x → C(a, b; x) are strictly increasing on [1/2, 1] and
Q(a, b; 1/2) = A(a, b) < T [A(a, b), Q(a, b)] < Q(a, b) = Q(a, b; 1),
(1.5)
C(a, b; 1/2) = A(a, b) < T [A(a, b), Q(a, b)] < Q(a, b) < C(a, b) = C(a, b; 1) (1.6)
for all a, b > 0 with a 6= b.
Motivated by inequalities (1.5) and (1.6), it is natural to ask what are the best
possible parameters λ1 , µ1 , λ2 , µ2 ∈ (1/2, 1) such that the double inequalities
Q(a, b; λ1 ) < T [A(a, b), Q(a, b)] < Q(a, b; µ1 ),
C(a, b; λ2 ) < T [A(a, b), Q(a, b)] < C(a, b; µ2 )
hold for all a, b > 0 with a 6= b? The main purpose of this paper is to answer this
question.
2. Lemmas
In order to prove our main results we need several lemmas, which we present in
this section.
Lemma 2.1. (See [5, Exercise 3.43(11)]) The double inequality
K(r) − E(r)
π
π
<
<
2
4
r
4(1 − r2 )
holds for all r ∈ (0, 1).
Lemma 2.2. (See [5, Exercise 3.43(16)]) The function r → [E 2 (r)−(1−r2 )K2 (r)]/r4
is strictly increasing from (0, 1) onto (π 2 /32, 1).
Lemma 2.3. (See [5, Theorem 1.25]) Let a, b ∈ R with a < b, f, g : [a, b] →
R be continuous on [a, b] and differentiable on (a, b) and g 0 (x) 6= 0 on (a, b). If
f 0 (x)/g 0 (x) is increasing (decreasing) on (a, b), then so are the functions
f (x) − f (a)
,
g(x) − g(a)
f (x) − f (b)
.
g(x) − g(b)
If f 0 (x)/g 0 (x) is strictly monotone, then the monotonicity in the conclusion is also
strict.
Lemma
2.4. The function
−√E(r)]/r2 is strictly increasing from
√
√ r → E(r)[K(r)
√
2
(0, 2/2) onto (π /8, 2E( 2/2)[K( 2/2) − E( 2/2)]).
Proof. Let
φ(r) =
E(r)[K(r) − E(r)]
.
r2
(2.1)
Then simple computations lead to
r E 2 (r) − (1 − r2 )K2 (r)
.
1 − r2
r4
It follows from Lemmas 2.1 and 2.2 together with (2.1) and (2.2) that
√ !
√ !"
√ !
√ !#
π2
2
2
2
2
+
, φ
φ(0 ) =
= 2E
K
−E
8
2
2
2
2
√
and φ(r) is strictly increasing on (0, 2/2).
φ0 (r) =
(2.2)
(2.3)
5
Wei-Mao Qian, Yu-Ming Chu and Ying-Qing Song
√
Lemma 2.5. The function√r → [r2 E(r) + (1√− r2 )(K(r)
− E(r))]/(r2 1 − r2 ) is
√
strictly increasing from (0, 2/2) onto (3π/4, 2K( 2/2)).
Proof. Let
ϕ1 (r) = r2 E(r) + (1 − r2 )(K(r) − E(r)), ϕ2 (r) = r2
p
ϕ1 (r)
1 − r2 , ϕ(r) =
. (2.4)
ϕ2 (r)
Then elaborated computations give
ϕ1 (0+ ) = ϕ2 (0+ ) = 0,
√ !
√ !
√
2
2
ϕ
= 2K
,
2
2
ϕ01 (r)
= 3ψ(r),
ϕ02 (r)
where
(2.5)
(2.6)
(2.7)
√
1 − r2 [2E(r) − K(r)]
,
2 − 3r2
π
ψ(0+ ) = ,
4
ω(r)
,
ψ 0 (r) = √
r 1 − r2 (2 − 3r2 )2
ψ(r) =
(2.8)
(2.9)
where
ω(r) = 2[E(r) − K(r)] + r2 [E(r) + K(r)],
ω 0 (r) =
ω(0+ ) = 0,
(2.10)
r(2 − 3r2 )E(r)
>0
1 − r2
(2.11)
√
for r ∈ (0, 2/2).
0
0
It
√ follows from (2.7) and (2.9)-(2.11) that φ1 (r)/φ2 (r) is strictly increasing on
(0, 2/2). Then (2.4) and (2.5) together
with Lemma 2.3 lead to the conclusion
√
that ϕ(r) is strictly increasing on (0, 2/2).
From (2.5), (2.7) and (2.8) we clearly see that
3π
.
(2.12)
4
Therefore, Lemma 2.5 follows from
√ (2.6) and (2.12) together with the monotonicity of ϕ(r) on the interval (0, 2/2).
ϕ(0+ ) =
3. Main Results
Theorem 3.1. Let λ1 , µ1 ∈ (1/2, 1). Then the double inequality
Q(a, b; λ1 ) < T [A(a, b), Q(a, b)] < Q(a, b; µ1 )
q
√
holds for all a, b > 0 with a 6= b if and only if λ1 ≤ 1/2 + 2E 2 ( 2/2)/π 2 − 1/4 =
√
0.8459 · · · and µ1 ≥ 1/2 + 2/4 = 0.8535 · · · .
Optimal Bounds for a Toader-Type Mean in Terms of One-Parameter Quadratic and
Contraharmonic Means
6
Proof. Sine Q(a, b), T (a, b) and A(a, b) are symmetric and homogeneous
p of degree 1,
without
loss
of
generality,
we
assume
that
a
>
b
>
0.
Let
r
=
(a−b)/
2(a2 + b2 ) ∈
√
(0, 2/2) and p ∈ (1/2, 1). Then (1.1), (1.2) and (1.4) lead to
2A(a, b)
√
E(r),
π 1 − r2
A(a, b) p
Q(a, b; p) = √
1 − 4p(1 − p)r2 .
1 − r2
It follows from (3.1), (3.2) and Lemma 2.4 that
T [A(a, b), Q(a, b)] =
Q(a, b; p) − T [A(a, b), Q(a, b)] = √
(3.2)
A(a, b)
1−
r2
hp
i F (r), (3.3)
1 − 4p(1 − p)r2 + π2 E(r)
where
F (r) = 1 − 4p(1 − p)r2 −
F
(3.1)
4 2
E (r),
π2
F (0+ ) = 0,
!
√
4
2
= 1 − 2p(1 − p)r2 − 2 E 2
2
π
(3.4)
√ !
2
,
2
F 0 (r) = 8rf (r),
(3.5)
(3.6)
where
E(r)[K(r) − E(r)]
− p(1 − p),
π2 r2
1
f (0+ ) = − p(1 − p),
8
√ h √ √ i
√ ! 2E
2
K 22 − E 22
2
2
=
− p(1 − p).
2
π2
f (r) =
f
(3.7)
(3.8)
(3.9)
We divide the proof into four
q cases.
√
Case 1.1 p = p0 = 1/2 + 2E 2 ( 2/2)/π 2 − 1/4. Then (3.5), (3.8) and (3.9)
lead to
√ !
2
F
= 0,
(3.10)
2
√ !
2 2
2
3
+
− = −0.005331 · · · < 0,
(3.11)
f (0 ) = 2 E
π
2
8
√ !
√ !
√ !
2
2
2
2
1
f
= 2E
K
− = 0.007455 · · · > 0.
(3.12)
2
π
2
2
2
From√Lemma 2.4, (3.6), (3.7), (3.11) and (3.12) we clearly see that there exists
r0 ∈ (0,√ 2/2) such that F (r) is strictly decreasing on (0, r0 ) and strictly increasing
on (r0 , 2/2). Therefore, T [A(a, b), Q(a, b)] > Q(a, b; p0 ) follows from (3.3), (3.4)
and (3.10) together with the√piecewise monotonicity of F (r).
Case 1.2 p = p1 = 1/2 + 2/4. Then (3.8) becomes
f (0+ ) = 0.
(3.13)
7
Wei-Mao Qian, Yu-Ming Chu and Ying-Qing Song
It follows
from Lemma 2.4, (3.6), (3.7) and (3.13) that F (r) is strictly increasing
√
on (0, 2/2). Therefore, T [A(a, b), Q(a, b)] < Q(a, b; p1 ) follows easily from (3.3)
and (3.4) together q
with the monotonicity of F (r).
√
Case 1.3 1/2 + 2E 2 ( 2/2)/π 2 − 1/4 < p = p2 < 1. Then (3.5) leads to
√ !
2
F
> 0.
(3.14)
2
Equation
(3.3) and inequality (3.14) imply that there exists small enough δ1 ∈
√
Q(a, b)] < Q(a, b; p2 ) for all a > b > 0 with (a −
(0, p2/2) such that√T [A(a, b),√
b)/ 2(a2 + b2 ) ∈ ( 2/2 − δ1 , 2/2).
√
Case 1.4 1/2 < p = p3 < 1/2 + 2/4. Then (3.8) leads to
f (0+ ) < 0.
(3.15)
Equations (3.3), (3.4) and (3.6) together with
√ inequality (3.15) leads to the conclusion that there exists small enough δ2 ∈
(0,
2/2) such that T [A(a, b), Q(a, b)] >
p
Q(a, b; p3 ) for all a > b > 0 with (a − b)/ 2(a2 + b2 ) ∈ (0, δ2 ).
Theorem 3.2. Let λ2 , µ2 ∈ (1/2, 1). Then the double inequality
C(a, b; λ2 ) < T [A(a, b), Q(a, b)] < C(a, b; µ2 )
q√
√
holds for all a, b > 0 with a 6= b if and only if λ2 ≤ 1/2+
2E( 2/2)/(2π) − 1/4 =
0.7323 · · · and µ2 ≥ 3/4.
Proof.
of generality, we assume that a > b > 0. Let r = (a −
p Without loss √
b)/ 2(a2 + b2 ) ∈ (0, 2/2) and q ∈ (1/2, 1). Then (1.1), (1.2) and (1.4) lead to
A(a, b)
[1 − 4q(1 − q)r2 ].
1 − r2
It follows from (3.1), (3.16) and Lemma 2.5 that
C(a, b; q) =
C(a, b; q) − T [A(a, b), Q(a, b)] =
where
A(a, b)
G(r),
1 − r2
2p
1 − r2 E(r),
π
G(0+ ) = 0,
√ !
√
√ !
2
2
2
= 1 − 2q(1 − q) −
E
,
2
π
2
(3.16)
(3.17)
G(r) = 1 − 4q(1 − q)r2 −
G
G0 (r) = 2rg(r),
where
r2 E(r) + (1 − r2 )[K(r) − E(r)]
√
− 4q(1 − q),
πr2 1 − r2
3
g(0+ ) = − 4q(1 − q),
4
√ ! √
√ !
2
2
2
g
=
K
− 4q(1 − q).
2
π
2
g(r) =
We divide the proof into four cases.
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
Optimal Bounds for a Toader-Type Mean in Terms of One-Parameter Quadratic and
Contraharmonic Means
Case 2.1 q = q0 = 1/2 +
(3.23) lead to
g
q√
8
√
2E( 2/2)/(2π) − 1/4. Then (3.19), (3.22) and
√ !
2
G
= 0,
2
√ √ 2 2E 22
5
g(0+ ) =
− = −0.03399 · · · < 0,
π
4
h √ √ i
√
!
√
2
2
2
2E
+
K
2
2
2
=
− 2 = 0.05063 · · · .
2
π
(3.24)
(3.25)
(3.26)
From Lemma
√ 2.5, (3.20), (3.21), (3.25) and (3.26) we clearly ∗see that there
exists r∗ ∈ (0, 2/2)
√ such that G(r) is strictly decreasing on (0, r ) and strictly
∗
increasing on (r , 2/2). Therefore, T [A(a, b), Q(a, b)] > C(a, b; q0 ) follows from
(3.17), (3.18), (3.24) and the piecewise monotonicity of G(r).
Case 2.2 q = q1 = 3/4. Then (3.22) becomes
g(0+ ) = 0.
(3.27)
It follows
√ from Lemma 2.5, (3.20), (3.21) and (3.27) that G(r) is strictly increasing on (0, 2/2). Therefore, T [A(a, b), Q(a, b)] < C(a, b; q1 ) follows from (3.17) and
(3.18) together with
monotonicity of G(r).
qthe
√
√
Case 2.3 1/2 +
2E( 2/2)/(2π) − 1/4 < q = q3 < 1. Then (3.19) leads to
√ !
2
G
> 0.
(3.28)
2
Equation
(3.17) and inequality (3.28) imply that there exists small enough δ3 ∈
√
(0, p2/2) such that√T [A(a, b),√Q(a, b)] < C(a, b; q3 ) for all a > b > 0 with (a −
b)/ 2(a2 + b2 ) ∈ ( 2/2 − δ3 , 2/2).
Case 2.4 1/2 < q = q4 < 3/4. Then (3.22) leads to
g(0+ ) < 0.
(3.29)
Equations (3.17), (3.18) and (3.20)
√ together with inequality (3.29) imply that
there exists small enough δ4 ∈p(0, 2/2) such that T [A(a, b), Q(a, b)] > C(a, b; q4 )
for all a > b > 0 with (a − b)/ 2(a2 + b2 ) ∈ (0, δ4 ).
From Theorems 3.1 and 3.2 we get Corollary 3.3 immediately.
q
√
√
Corollary 3.3. Let λ1 = 1/2 + 2E 2 ( 2/2)/π 2 − 1/4, µ1 = 1/2 + 2/4, λ2 =
q√
√
1/2 +
2E( 2/2)/(2π) − 1/4 and µ2 = 3/4. Then the double inequality
p
1 − 4λ2 (1 − λ2 )r2
π
2
√
max
1 − 4λ1 (1 − λ1 )r ,
< E(r)
2
1 − r2
p
π
1 − 4µ2 (1 − µ2 )r2
2
√
< min
1 − 4µ1 (1 − µ1 )r ,
2
1 − r2
√
for all r ∈ (0, 2/2).
9
Wei-Mao Qian, Yu-Ming Chu and Ying-Qing Song
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Wei-Mao Qian, School of Distance Education, Huzhou Broadcast and TV University,
Huzhou 313000, China
E-mail address: qwm661977@126.com
Yu-Ming Chu (Corresponding author), School of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China
E-mail address: chuyuming2005@126.com
Ying-Qing Song, School of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China
E-mail address: 1452225875@qq.com