# Sharp Bounds for Neuman Means by Harmonic, Arithmetic, and Contraharmonic Means

Authors: Zhi-Jun Guo; Yu-Ming Chu; Ying-Qing Song; Xiao-Jing Tao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We give several sharp bounds for the Neuman meansN
A
H and N
H
A (N
C
A and N
A
C) in terms of harmonic
mean H (contraharmonic mean C) or the geometric convex
combination of arithmetic mean A and harmonic mean H
(contraharmonic mean C and arithmetic mean A) and present a
new chain of inequalities for certain bivariate means.


---

## Body

## 1. Introduction

Fora
,
b
>
0 with a
≠
b, the Schwab-Borchardt mean SB
(
a
,
b
) [1–3] of a and b is defined as
(1)
SB
(
a
,
b
)
=
{
b
2
-
a
2
cos
⁡
-
1
(
a
/
b
)
,
a
<
b
,
a
2
-
b
2
cosh
⁡
-
1
(
a
/
b
)
,
a
>
b
,
where cos
⁡
-
1
(
x
) and cosh
⁡
-
1
(
x
)
=
log
⁡
⁡
(
x
+
x
2
-
1
) are the inverse cosine and inverse hyperbolic cosine functions, respectively.The Schwab-Borchardt meanSB
(
a
,
b
) can be expressed by the symmetric elliptic integral R
F [4] of the first kind as follows [5] (see also [6, (3.21)]):
(2)
SB
(
a
,
b
)
=
1
R
F
(
a
2
,
b
2
,
b
2
)
,
where R
F
(
a
,
b
,
c
)
=
(
1
/
2
)
∫
0
∞
[
(
t
+
a
)
(
t
+
b
)
(
t
+
c
)
]
-
1
/
2
d
t.Recently, the Schwab-Borchardt mean has been the subject of intensive research. In particular, many remarkable inequalities for the Schwab-Borchardt mean and its generated means can be found in the literature [1–3, 7–10].Very recently, Neuman [11] found a new mean N
(
a
,
b
) derived from the Schwab-Borchardt mean as follows:
(3)
N
(
a
,
b
)
=
1
2
[
a
+
b
2
SB
(
a
,
b
)
]
.LetG
(
a
,
b
)
=
a
b, H
(
a
,
b
)
=
2
a
b
/
(
a
+
b
), L
(
a
,
b
)
=
(
a
-
b
)
/
(
log
⁡
⁡
a
-
log
⁡
⁡
b
), P
(
a
,
b
)
=
(
a
-
b
)
/
[
2
sin
-
1
(
(
a
-
b
)
/
(
a
+
b
)
)
], A
(
a
,
b
)
=
(
a
+
b
)
/
2, M
(
a
,
b
)
=
(
a
-
b
)
/
[
2
sinh
⁡
-
1
(
(
a
-
b
)
/
(
a
+
b
)
)
], T
(
a
,
b
)
=
(
a
-
b
)
/
[
2
tan
-
1
(
(
a
-
b
)
/
(
a
+
b
)
)
], Q
(
a
,
b
)
=
(
a
2
+
b
2
)
/
2, and C
(
a
,
b
)
=
(
a
2
+
b
2
)
/
(
a
+
b
) be, respectively, the geometric, harmonic, logarithmic, first Seiffert, arithmetic, Neuman-Sándor, second Seiffert, quadratic, and contraharmonic means, and let
(4)
N
A
H
(
a
,
b
)
=
N
[
A
(
a
,
b
)
,
H
(
a
,
b
)
]
,
N
H
A
(
a
,
b
)
=
N
[
H
(
a
,
b
)
,
A
(
a
,
b
)
]
,
N
A
G
(
a
,
b
)
=
N
[
A
(
a
,
b
)
,
G
(
a
,
b
)
]
,
N
G
A
(
a
,
b
)
=
N
[
G
(
a
,
b
)
,
A
(
a
,
b
)
]
,
N
A
C
(
a
,
b
)
=
N
[
A
(
a
,
b
)
,
C
(
a
,
b
)
]
,
N
C
A
(
a
,
b
)
=
N
[
C
(
a
,
b
)
,
A
(
a
,
b
)
]
,
N
A
Q
(
a
,
b
)
=
N
[
A
(
a
,
b
)
,
Q
(
a
,
b
)
]
,
N
Q
A
(
a
,
b
)
=
N
[
Q
(
a
,
b
)
,
A
(
a
,
b
)
]
be the Neuman means. Then Neuman [11] proved that
(5)
G
(
a
,
b
)
<
L
(
a
,
b
)
<
N
A
G
(
a
,
b
)
<
P
(
a
,
b
)
<
N
G
A
(
a
,
b
)
<
A
(
a
,
b
)
<
M
(
a
,
b
)
<
N
Q
A
(
a
,
b
)
<
T
(
a
,
b
)
<
N
A
Q
(
a
,
b
)
<
Q
(
a
,
b
)
<
C
(
a
,
b
)
,
for all a
,
b
>
0 with a
≠
b, and the double inequalities
(6)
α
1
A
(
a
,
b
)
+
(
1
-
α
1
)
G
(
a
,
b
)
<
N
G
A
(
a
,
b
)
<
β
1
A
(
a
,
b
)
+
(
1
-
β
1
)
G
(
a
,
b
)
,
α
2
Q
(
a
,
b
)
+
(
1
-
α
2
)
A
(
a
,
b
)
<
N
A
Q
(
a
,
b
)
<
β
2
Q
(
a
,
b
)
+
(
1
-
β
2
)
A
(
a
,
b
)
,
α
3
A
(
a
,
b
)
+
(
1
-
α
3
)
G
(
a
,
b
)
<
N
A
G
(
a
,
b
)
<
β
3
A
(
a
,
b
)
+
(
1
-
β
3
)
G
(
a
,
b
)
,
α
4
Q
(
a
,
b
)
+
(
1
-
α
4
)
A
(
a
,
b
)
<
N
Q
A
(
a
,
b
)
<
β
4
Q
(
a
,
b
)
+
(
1
-
β
4
)
A
(
a
,
b
)
hold for all a
,
b
>
0 with a
≠
b if and only if α
1
≤
2
/
3, β
1
≥
π
/
4, α
2
≤
2
/
3, β
2
≥
(
π
-
2
)
/
[
4
(
2
-
1
)
]
=
0.689
…, α
3
≤
1
/
3, β
3
≥
1
/
2, α
4
≤
1
/
3, and β
4
≥
[
log
⁡
(
1
+
2
)
+
2
-
2
]
/
[
2
(
2
-
1
)
]
=
0.356
….Zhang et al. [12] presented the best possible parameters α
1
,
α
2
,
β
1
,
β
2
∈
[
0,1
/
2
] and α
3
,
α
4
,
β
3
,
β
4
∈
[
1
/
2,1
] such that the double inequalities
(7)
G
(
α
1
a
+
(
1
-
α
1
)
b
,
α
1
b
+
(
1
-
α
1
)
a
)
<
N
A
G
(
a
,
b
)
<
G
(
β
1
a
+
(
1
-
β
1
)
b
,
β
1
b
+
(
1
-
β
1
)
a
)
,
G
(
α
2
a
+
(
1
-
α
2
)
b
,
α
2
b
+
(
1
-
α
2
)
a
)
<
N
G
A
(
a
,
b
)
<
G
(
β
2
a
+
(
1
-
β
2
)
b
,
β
2
b
+
(
1
-
β
2
)
a
)
,
Q
(
α
3
a
+
(
1
-
α
3
)
b
,
α
3
b
+
(
1
-
α
3
)
a
)
<
N
Q
A
(
a
,
b
)
<
Q
(
β
3
a
+
(
1
-
β
3
)
b
,
β
3
b
+
(
1
-
β
3
)
a
)
,
Q
(
α
4
a
+
(
1
-
α
4
)
b
,
α
4
b
+
(
1
-
α
4
)
a
)
<
N
A
Q
(
a
,
b
)
<
Q
(
β
4
a
+
(
1
-
β
4
)
b
,
β
4
b
+
(
1
-
β
4
)
a
)
hold for all a
,
b
>
0 with a
≠
b.In [13], the authors found the greatest values α
1, α
2, α
3, α
4, α
5, α
6, α
7, and α
8 and the least values β
1, β
2, β
3, β
4, β
5, β
6, β
7, and β
8 such that the double inequalities
(8)
A
α
1
(
a
,
b
)
G
1
-
α
1
(
a
,
b
)
<
N
G
A
(
a
,
b
)
<
A
β
1
(
a
,
b
)
G
1
-
β
1
(
a
,
b
)
,
α
2
G
(
a
,
b
)
+
1
-
α
2
A
(
a
,
b
)
<
1
N
G
A
(
a
,
b
)
<
β
2
G
(
a
,
b
)
+
1
-
β
2
A
(
a
,
b
)
,
A
α
3
(
a
,
b
)
G
1
-
α
3
(
a
,
b
)
<
N
A
G
(
a
,
b
)
<
A
β
3
(
a
,
b
)
G
1
-
β
3
(
a
,
b
)
,
α
4
G
(
a
,
b
)
+
1
-
α
4
A
(
a
,
b
)
<
1
N
A
G
(
a
,
b
)
<
β
4
G
(
a
,
b
)
+
1
-
β
4
A
(
a
,
b
)
,
Q
α
5
(
a
,
b
)
A
1
-
α
5
(
a
,
b
)
<
N
A
Q
(
a
,
b
)
<
Q
β
5
(
a
,
b
)
A
1
-
β
5
(
a
,
b
)
,
α
6
A
(
a
,
b
)
+
1
-
α
6
Q
(
a
,
b
)
<
1
N
A
Q
(
a
,
b
)
<
β
6
A
(
a
,
b
)
+
1
-
β
6
Q
(
a
,
b
)
,
Q
α
7
(
a
,
b
)
A
1
-
α
7
(
a
,
b
)
<
N
Q
A
(
a
,
b
)
<
Q
β
7
(
a
,
b
)
A
1
-
β
7
(
a
,
b
)
,
α
8
A
(
a
,
b
)
+
1
-
α
8
Q
(
a
,
b
)
<
1
N
Q
A
(
a
,
b
)
<
β
8
A
(
a
,
b
)
+
1
-
β
8
Q
(
a
,
b
)
hold for all a
,
b
>
0 with a
≠
b.Leta
>
b
>
0, v
=
(
a
-
b
)
/
(
a
+
b
)
∈
(
0,1
), p
∈
(
0
,
∞
), q
∈
(
0
,
π
/
2
), r
∈
(
0
,
log
⁡
(
2
+
3
)
), and s
∈
(
0
,
π
/
3
) be the parameters such that 1
/
cosh
⁡
(
p
)
=
cos
⁡
(
q
)
=
1
-
v
2, cosh
⁡
(
r
)
=
sec
(
s
)
=
1
+
v
2. Then He et al. [14] proved that
(9)
N
A
H
(
a
,
b
)
=
1
2
A
(
a
,
b
)
[
1
+
2
p
sinh
⁡
(
2
p
)
]
,
(10)
N
H
A
(
a
,
b
)
=
1
2
A
(
a
,
b
)
[
cos
⁡
(
q
)
+
q
sin
(
q
)
]
,
(11)
N
C
A
(
a
,
b
)
=
1
2
A
(
a
,
b
)
[
cosh
⁡
(
r
)
+
r
sinh
⁡
(
r
)
]
,
(12)
N
A
C
(
a
,
b
)
=
1
2
A
(
a
,
b
)
[
1
+
2
s
sin
⁡
(
2
s
)
]
,
(13)
H
(
a
,
b
)
<
N
A
H
(
a
,
b
)
<
N
H
A
(
a
,
b
)
<
A
(
a
,
b
)
<
N
C
A
(
a
,
b
)
<
N
A
C
(
a
,
b
)
<
C
(
a
,
b
)
.Letx
∈
[
0,1
/
2
], y
∈
[
1
/
2,1
], f
(
x
)
=
H
[
x
a
+
(
1
-
x
)
b
,
x
b
+
(
1
-
x
)
a
], and g
(
y
)
=
C
[
y
a
+
(
1
-
y
)
b
,
y
b
+
(
1
-
y
)
a
]. Then we clearly see that
(14)
f
(
0
)
=
H
(
a
,
b
)
<
N
A
H
(
a
,
b
)
<
N
H
A
(
a
,
b
)
<
A
(
a
,
b
)
=
f
(
1
2
)
=
g
(
1
2
)
<
N
C
A
(
a
,
b
)
<
N
A
C
(
a
,
b
)
<
C
(
a
,
b
)
=
g
(
1
)
,
for all a
,
b
>
0 with a
≠
b, and the functions f and g are, respectively, strictly increasing on the intervals [
0,1
/
2
] and [
1
/
2,1
] for fixed a
,
b
>
0 with a
≠
b.The main purpose of this paper is to find the best possible parametersα
1, α
2, α
3, α
4, β
1, β
2, β
3, β
4
∈
R, α
5
,
α
6
,
β
5
,
β
6
∈
[
0,1
/
2
], and α
7
,
α
8
,
β
7
,
β
8
∈
[
1
/
2,1
] such that the double inequalities
(15)
A
α
1
(
a
,
b
)
H
1
-
α
1
(
a
,
b
)
<
N
A
H
(
a
,
b
)
<
A
β
1
(
a
,
b
)
H
1
-
β
1
(
a
,
b
)
,
A
α
2
(
a
,
b
)
H
1
-
α
2
(
a
,
b
)
<
N
H
A
(
a
,
b
)
<
A
β
2
(
a
,
b
)
H
1
-
β
2
(
a
,
b
)
,
C
α
3
(
a
,
b
)
A
1
-
α
3
(
a
,
b
)
<
N
C
A
(
a
,
b
)
<
C
β
3
(
a
,
b
)
A
1
-
β
3
(
a
,
b
)
,
C
α
4
(
a
,
b
)
A
1
-
α
4
(
a
,
b
)
<
N
A
C
(
a
,
b
)
<
C
β
4
(
a
,
b
)
A
1
-
β
4
(
a
,
b
)
,
H
[
α
5
a
+
(
1
-
α
5
)
b
,
α
5
b
+
(
1
-
α
5
)
a
]
<
N
A
H
(
a
,
b
)
<
H
[
β
5
a
+
(
1
-
β
5
)
b
,
β
5
b
+
(
1
-
β
5
)
a
]
,
H
[
α
6
a
+
(
1
-
α
6
)
b
,
α
6
b
+
(
1
-
α
6
)
a
]
<
N
H
A
(
a
,
b
)
<
H
[
β
6
a
+
(
1
-
β
6
)
b
,
β
6
b
+
(
1
-
β
6
)
a
]
,
C
[
α
7
a
+
(
1
-
α
7
)
b
,
α
7
b
+
(
1
-
α
7
)
a
]
<
N
C
A
(
a
,
b
)
<
H
[
β
7
a
+
(
1
-
β
7
)
b
,
β
7
b
+
(
1
-
β
7
)
a
]
,
C
[
α
8
a
+
(
1
-
α
8
)
b
,
α
8
b
+
(
1
-
α
8
)
a
]
<
N
A
C
(
a
,
b
)
<
H
[
β
8
a
+
(
1
-
β
8
)
b
,
β
8
b
+
(
1
-
β
8
)
a
]
hold, for all a
,
b
>
0 with a
≠
b, and present a new chain of inequalities for certain bivariate means.


## 2. Lemmas

In order to prove our main results we need several lemmas, which we present in this section.Lemma 1 (see [15, Theorem 1.25]).
For-
∞
<
a
<
b
<
∞, let f
,
g
:
[
a
,
b
]
→
R be continuous on [
a
,
b
] and differentiable on (
a
,
b
); let g
′
(
x
)
≠
0 on (
a
,
b
). If f
′
(
x
)
/
g
′
(
x
) is increasing (decreasing) on (
a
,
b
), then so are
(16)
f
(
x
)
-
f
(
a
)
g
(
x
)
-
g
(
a
)
,
f
(
x
)
-
f
(
b
)
g
(
x
)
-
g
(
b
)
.
If f
′
(
x
)
/
g
′
(
x
) is strictly monotone, then the monotonicity in the conclusion is also strict.Lemma 2 (see [16, Lemma 1.1]).
Suppose that the power seriesf
(
x
)
=
∑
n
=
0
∞
‍
a
n
x
n and g
(
x
)
=
∑
n
=
0
∞
‍
b
n
x
n have the radius of convergence r
>
0 and a
n
,
b
n
>
0 for all n
≥
0. If the sequence {
a
n
/
b
n
} is (strictly) increasing (decreasing) for all n
≥
0, then the function f
(
x
)
/
g
(
x
) is also (strictly) increasing (decreasing) on (
0
,
r
).Lemma 3.
The function(17)
φ
1
(
x
)
=
sinh
⁡
(
x
)
-
x
sinh
⁡
(
x
)
[
cosh
⁡
(
x
)
-
1
]
is strictly decreasing from (
0
,
∞
) onto (
0,1
/
3
).Proof.
Making use of power series expansion we get(18)
φ
1
(
x
)
=
2
sinh
⁡
⁡
(
x
)
-
2
x
sinh
⁡
⁡
(
2
x
)
-
2
sinh
⁡
⁡
(
x
)
=
∑
n
=
1
∞
(
2
/
(
2
n
+
1
)
!
)
x
2
n
+
1
∑
n
=
1
∞
(
(
2
2
n
+
1
-
2
)
/
(
2
n
+
1
)
!
)
x
2
n
+
1
=
∑
n
=
0
∞
(
2
/
(
2
n
+
3
)
!
)
x
2
n
∑
n
=
0
∞
(
(
2
2
n
+
3
-
2
)
/
(
2
n
+
3
)
!
)
x
2
n
.
Let(19)
a
n
=
2
(
2
n
+
3
)
!
,
b
n
=
2
2
n
+
3
-
2
(
2
n
+
3
)
!
.
Then
(20)
a
n
>
0
,
b
n
>
0
,
and a
n
/
b
n
=
1
/
(
2
2
n
+
2
-
1
) is strictly decreasing for all n
≥
0.
Note that(21)
φ
1
(
0
+
)
=
a
0
b
0
=
1
3
,
φ
1
(
∞
)
=
lim
⁡
n
→
∞
a
n
b
n
=
0
.
Therefore, Lemma3 follows easily from Lemma 2 and (18)–(21) together with the monotonicity of the sequence {
a
n
/
b
n
}.Lemma 4.
The function(22)
φ
2
(
x
)
=
x
-
sin
(
x
)
sin
(
x
)
[
1
-
cos
⁡
(
x
)
]
is strictly increasing from (
0
,
π
/
2
) onto (
1
/
3
,
π
/
2
-
1
).Proof.
Letf
1
(
x
)
=
x
-
sin
(
x
) and g
1
(
x
)
=
sin
(
x
)
[
1
-
cos
⁡
(
x
)
]. Then simple computations lead to
(23)
φ
2
(
x
)
=
f
1
(
x
)
g
1
(
x
)
=
f
1
(
x
)
-
f
1
(
0
)
g
1
(
x
)
-
g
1
(
0
)
,
and f
1
′
(
x
)
/
g
1
′
(
x
)
=
1
/
[
1
+
2
cos
⁡
(
x
)
] is strictly increasing on (
0
,
π
/
2
).
Note that(24)
φ
2
(
0
+
)
=
lim
⁡
x
→
0
+
f
1
′
(
x
)
g
1
′
(
x
)
=
1
3
,
φ
2
(
π
2
)
=
π
2
-
1
.
Therefore, Lemma4 follows from Lemma 1, (23), (24), and the monotonicity of f
1
′
(
x
)
/
g
1
′
(
x
).Lemma 5.
The function(25)
φ
3
(
x
)
=
log
⁡
⁡
[
(
x
+
sinh
⁡
(
x
)
cosh
⁡
(
x
)
)
/
(
2
sinh
⁡
(
x
)
cosh
⁡
(
x
)
)
]
log
⁡
⁡
(
1
/
cosh
⁡
(
x
)
)
is strictly decreasing from (
0
,
∞
) onto (
0,2
/
3
).Proof.
Letf
2
(
x
)
=
log
⁡
[
(
x
+
sinh
⁡
(
x
)
cosh
⁡
(
x
)
)
/
(
2
sinh
⁡
(
x
)
cosh
⁡
(
x
)
)
] and g
2
(
x
)
=
log
⁡
[
1
/
cosh
⁡
(
x
)
]. Then simple computations lead to
(26)
φ
3
(
x
)
=
f
2
(
x
)
g
2
(
x
)
=
f
2
(
x
)
-
f
2
(
0
+
)
g
2
(
x
)
-
g
2
(
0
)
,
(27)
f
2
′
(
x
)
g
2
′
(
x
)
=
(
x
[
cosh
⁡
3
(
x
)
+
sinh
⁡
2
(
x
)
cosh
⁡
⁡
(
x
)
]
-
sinh
⁡
⁡
(
x
)
cosh
⁡
2
(
x
)
)
×
(
sinh
⁡
2
(
x
)
cosh
⁡
⁡
(
x
)
[
x
+
sinh
⁡
⁡
(
x
)
cosh
⁡
⁡
(
x
)
]
)
-
1
=
(
8
x
cosh
⁡
(
3
x
)
+
8
x
cosh
⁡
(
x
)
-
4
sinh
⁡
⁡
(
3
x
)
-
4
sinh
⁡
⁡
(
x
)
)
×
(
4
x
cosh
⁡
(
3
x
)
-
4
x
cosh
⁡
(
x
)
+
sinh
⁡
(
5
x
)
-
sinh
⁡
(
3
x
)
k
k
k
k
k
k
-
2
sinh
⁡
(
x
)
)
-
1
=
(
∑
n
=
1
∞
‍
(
(
(
2
n
+
1
)
!
)
-
1
(
8
(
2
n
+
1
)
(
1
+
3
2
n
)
-
4
(
1
+
3
2
n
+
1
)
)
K
K
K
K
K
K
×
(
(
2
n
+
1
)
!
)
-
1
)
x
2
n
+
1
∑
n
=
1
∞
‍
)
×
(
∑
n
=
1
∞
‍
(
(
(
2
n
+
1
)
!
)
-
1
(
4
(
2
n
+
1
)
(
3
2
n
-
1
)
+
5
2
n
+
1
-
3
2
n
+
1
-
2
)
K
K
K
K
K
K
K
×
(
(
2
n
+
1
)
!
)
-
1
)
x
2
n
+
1
∑
n
=
1
∞
‍
)
-
1
=
(
∑
n
=
0
∞
‍
(
(
(
2
n
+
3
)
!
)
-
1
(
8
(
2
n
+
3
)
(
1
+
3
2
n
+
2
)
-
4
(
1
+
3
2
n
+
3
)
)
k
k
k
k
k
k
k
k
×
(
(
2
n
+
3
)
!
)
-
1
)
x
2
n
∑
n
=
0
∞
‍
)
×
(
∑
n
=
0
∞
‍
(
(
(
2
n
+
3
)
!
)
-
1
(
4
(
2
n
+
3
)
(
3
2
n
+
2
-
1
)
+
5
2
n
+
3
-
3
2
n
+
3
-
2
)
k
k
k
k
k
k
k
k
k
k
×
(
(
2
n
+
3
)
!
)
-
1
)
x
2
n
∑
n
=
0
∞
‍
)
-
1
.
Let(28)
a
n
=
8
(
2
n
+
3
)
(
1
+
3
2
n
+
2
)
-
4
(
1
+
3
2
n
+
3
)
(
2
n
+
3
)
!
,
b
n
=
4
(
2
n
+
3
)
(
3
2
n
+
2
-
1
)
+
5
2
n
+
3
-
3
2
n
+
3
-
2
(
2
n
+
3
)
!
.
Then it is not difficult to verify that
(29)
a
n
=
(
16
n
+
20
)
+
(
16
n
+
12
)
3
2
n
+
2
(
2
n
+
3
)
!
>
0
,
b
n
>
0
,
a
n
+
1
b
n
+
1
-
a
n
b
n
=
-
c
n
d
n
<
0
for all n
≥
0, where c
n
=
(
256
n
+
48
)
3
2
n
+
2
×
5
2
n
+
3
+
(
384
n
+
464
)
5
2
n
+
3
+
(
2048
n
2
+
7040
n
+
6000
)
3
2
n
+
2
-
16
·
3
4
n
+
7
+
64 and d
n
=
[
5
2
n
+
5
+
(
8
n
+
17
)
3
2
n
+
4
-
(
8
n
+
22
)
]
[
5
2
n
+
3
+
(
8
n
+
9
)
3
2
n
+
2
-
(
8
n
+
14
)
].
Note that(30)
φ
3
(
0
+
)
=
a
0
b
0
=
2
3
,
φ
3
(
∞
)
=
0
.
It follows from Lemma2 and (27)–(29) that the function f
2
′
(
x
)
/
g
2
′
(
x
) is strictly decreasing on (
0
,
∞
). Therefore, Lemma 5 follows from (26) and (30) together with Lemma 1 and the monotonicity of f
2
′
(
x
)
/
g
2
′
(
x
).Lemma 6.
The function(31)
φ
4
(
x
)
=
log
⁡
[
(
x
+
sin
(
x
)
cos
⁡
(
x
)
)
/
(
2
sin
(
x
)
)
]
log
⁡
[
cos
⁡
(
x
)
]
is strictly decreasing from (
0
,
π
/
2
) onto (
0,1
/
3
).Proof.
Letf
3
(
x
)
=
log
⁡
[
(
x
+
sin
(
x
)
cos
⁡
(
x
)
)
/
(
2
sin
(
x
)
)
], g
3
(
x
)
=
log
⁡
[
cos
⁡
(
x
)
], f
4
(
x
)
=
cos
⁡
2
(
x
)
[
x
-
sin
(
x
)
cos
⁡
(
x
)
], and g
4
(
x
)
=
sin
2
(
x
)
[
x
+
sin
(
x
)
cos
⁡
(
x
)
]. Then simple computations lead to
(32)
φ
4
(
x
)
=
f
3
(
x
)
g
3
(
x
)
=
f
3
(
x
)
-
f
3
(
0
+
)
g
3
(
x
)
-
g
3
(
0
)
,
(33)
f
3
′
(
x
)
g
3
′
(
x
)
=
f
4
(
x
)
g
4
(
x
)
=
f
4
(
x
)
-
f
4
(
0
)
g
4
(
x
)
-
g
4
(
0
)
,
(34)
f
4
′
(
x
)
g
4
′
(
x
)
=
1
-
1
1
/
2
+
sin
(
2
x
)
/
(
2
x
)
.
Since the functionx
→
sin
(
x
)
/
x is strictly decreasing on (
0
,
π
), hence (34) leads to the conclusion that f
4
′
(
x
)
/
g
4
′
(
x
) is strictly decreasing on (
0
,
π
/
2
).
Note that(35)
φ
4
(
0
+
)
=
lim
⁡
x
→
0
+
f
4
′
(
x
)
g
4
′
(
x
)
=
1
3
,
φ
4
(
∞
)
=
0
.
Therefore, Lemma6 follows easily from (32), (33), (35), and Lemma 1 together with the monotonicity of f
4
′
(
x
)
/
g
4
′
(
x
).Lemma 7.
The function(36)
φ
5
(
x
)
=
log
⁡
(
(
x
+
sinh
⁡
(
x
)
cosh
⁡
(
x
)
)
/
(
2
sinh
⁡
(
x
)
)
)
log
⁡
[
cosh
⁡
(
x
)
]
is strictly increasing from (
0
,
log
⁡
(
2
+
3
)
) onto (
1
/
3
,
[
2
log
⁡
(
2
3
+
log
⁡
(
2
+
3
)
)
-
log
⁡
3
]
/
(
2
log
⁡
2
)
-
1
).Proof.
Letf
5
(
x
)
=
log
⁡
[
(
x
+
sinh
⁡
(
x
)
cosh
⁡
(
x
)
)
/
(
2
sinh
⁡
(
x
)
)
] and g
5
(
x
)
=
log
⁡
[
cosh
⁡
(
x
)
]. Then simple computations lead to
(37)
φ
5
(
x
)
=
f
5
(
x
)
g
5
(
x
)
=
f
5
(
x
)
-
f
5
(
0
+
)
g
5
(
x
)
-
g
5
(
0
)
,
(38)
f
5
′
(
x
)
g
5
′
(
x
)
=
sinh
⁡
(
x
)
cosh
⁡
3
(
x
)
-
x
cosh
⁡
2
(
x
)
x
sinh
⁡
2
(
x
)
+
sinh
⁡
3
(
x
)
cosh
⁡
(
x
)
=
sinh
⁡
(
4
x
)
+
2
sinh
⁡
(
2
x
)
-
4
x
cosh
⁡
(
2
x
)
-
4
x
sinh
⁡
(
4
x
)
-
2
sinh
⁡
(
2
x
)
+
4
x
cosh
⁡
(
2
x
)
-
4
x
=
∑
n
=
1
∞
(
2
2
n
+
2
(
2
2
n
-
2
n
)
/
(
2
n
+
1
)
!
)
x
2
n
+
1
∑
n
=
1
∞
(
2
2
n
+
2
(
2
2
n
+
2
n
)
/
(
2
n
+
1
)
!
)
x
2
n
+
1
=
∑
n
=
0
∞
(
2
2
n
+
4
[
2
2
n
+
2
-
(
2
n
+
2
)
]
/
(
2
n
+
3
)
!
)
x
2
n
∑
n
=
0
∞
(
2
2
n
+
4
[
2
2
n
+
2
+
(
2
n
+
2
)
]
/
(
2
n
+
3
)
!
)
x
2
n
.
Let(39)
a
n
=
2
2
n
+
4
[
2
2
n
+
2
-
(
2
n
+
2
)
]
(
2
n
+
3
)
!
,
b
n
=
2
2
n
+
4
[
2
2
n
+
2
+
(
2
n
+
2
)
]
(
2
n
+
3
)
!
.
Then(40)
a
n
>
0
,
b
n
>
0
,
a
n
+
1
b
n
+
1
-
a
n
b
n
=
(
3
n
+
2
)
2
2
n
+
2
(
2
2
n
+
1
+
n
+
1
)
(
2
2
n
+
3
+
n
+
2
)
>
0
for all n
≥
0.
It follows from Lemma2 and (38)–(40) that f
5
′
(
x
)
/
g
5
′
(
x
) is strictly increasing on (
0
,
log
⁡
(
2
+
3
)
).
Note that(41)
φ
5
(
0
+
)
=
a
0
b
0
=
1
3
φ
5
[
log
⁡
(
2
+
3
)
]
=
2
log
⁡
[
2
3
+
log
⁡
(
2
+
3
)
]
-
log
⁡
3
2
log
⁡
2
-
1
.
Therefore, Lemma7 follows from Lemma 1, (37), and (41) together with the monotonicity of f
5
′
(
x
)
/
g
5
′
(
x
).Lemma 8.
The function(42)
φ
6
(
x
)
=
log
⁡
[
(
x
+
sin
(
x
)
cos
⁡
(
x
)
)
/
(
2
sin
(
x
)
cos
⁡
(
x
)
)
]
log
⁡
⁡
[
sec
(
x
)
]
is strictly increasing from (
0
,
π
/
3
) onto (
2
/
3
,
[
2
log
⁡
(
4
π
+
3
3
)
-
2
log
⁡
6
-
log
⁡
3
]
/
(
2
log
⁡
2
)
).Proof.
Letf
6
(
x
)
=
log
⁡
[
(
x
+
sin
(
x
)
cos
⁡
(
x
)
)
/
(
2
sin
(
x
)
cos
⁡
(
x
)
)
], g
6
(
x
)
=
log
⁡
[
sec
(
x
)
], f
7
(
x
)
=
sin
(
x
)
cos
⁡
(
x
)
-
x
cos
⁡
(
2
x
), and g
7
(
x
)
=
(
x
+
sin
(
x
)
cos
⁡
(
x
)
)
sin
2
(
x
). Then simple computations lead to
(43)
φ
6
(
x
)
=
f
6
(
x
)
g
6
(
x
)
=
f
6
(
x
)
-
f
6
(
0
+
)
g
6
(
x
)
-
g
6
(
0
)
,
f
6
′
(
x
)
g
6
′
(
x
)
=
f
7
(
x
)
g
7
(
x
)
=
f
7
(
x
)
-
f
7
(
0
)
g
7
(
x
)
-
g
7
(
0
)
,
f
7
′
(
x
)
g
7
′
(
x
)
=
1
1
/
2
+
sin
(
2
x
)
/
2
x
.
Since the functionx
→
sin
(
2
x
)
/
(
2
x
) is strictly decreasing on (
0
,
π
/
3
), hence Lemma 1 and (43) lead to the conclusion that φ
6
(
x
) is strictly increasing on (
0
,
π
/
3
).
Note that(44)
φ
6
(
0
+
)
=
lim
⁡
x
→
0
+
f
7
′
(
x
)
g
7
′
(
x
)
=
2
3
,
(45)
φ
6
(
π
3
)
=
2
log
⁡
(
4
π
+
3
3
)
-
2
log
⁡
6
-
log
⁡
3
2
log
⁡
2
.
Therefore, Lemma8 follows easily from (44) and (45) together with the monotonicity of φ
6
(
x
).


## 3. Main Results

Theorem 9.
The double inequalities(46)
A
α
1
(
a
,
b
)
H
1
-
α
1
(
a
,
b
)
<
N
A
H
(
a
,
b
)
<
A
β
1
(
a
,
b
)
H
1
-
β
1
(
a
,
b
)
,
(47)
A
α
2
(
a
,
b
)
H
1
-
α
2
(
a
,
b
)
<
N
H
A
(
a
,
b
)
<
A
β
2
(
a
,
b
)
H
1
-
β
2
(
a
,
b
)
,
(48)
C
α
3
(
a
,
b
)
A
1
-
α
3
(
a
,
b
)
<
N
C
A
(
a
,
b
)
<
C
β
3
(
a
,
b
)
A
1
-
β
3
(
a
,
b
)
,
(49)
C
α
4
(
a
,
b
)
A
1
-
α
4
(
a
,
b
)
<
N
A
C
(
a
,
b
)
<
C
β
4
(
a
,
b
)
A
1
-
β
4
(
a
,
b
)
hold for all a
,
b
>
0 with a
≠
b if and only if α
1
≤
1
/
3, β
1
≥
1, α
2
≤
2
/
3, β
2
≥
1, α
3
≤
1
/
3, β
3
≥
[
2
log
⁡
(
2
3
+
log
⁡
(
2
+
3
)
)
-
log
⁡
3
]
/
(
2
log
⁡
2
)
-
1
=
0.4648
…, α
4
≤
2
/
3, and β
4
≥
[
2
log
⁡
(
4
π
+
3
3
)
-
2
log
⁡
6
-
log
⁡
3
]
/
(
2
log
⁡
2
)
=
0.7733
….Proof.
Without loss of generality, we assume thata
>
b
>
0. Let v
=
(
a
-
b
)
/
(
a
+
b
)
∈
(
0,1
), p
∈
(
0
,
∞
), q
∈
(
0
,
π
/
2
), r
∈
(
0
,
log
⁡
(
2
+
3
)
), and s
∈
(
0
,
π
/
3
) be the parameters such that 1
/
cosh
⁡
(
p
)
=
cos
⁡
(
q
)
=
1
-
v
2, cosh
⁡
(
r
)
=
sec
(
s
)
=
1
+
v
2. Then (9)–(12) lead to the conclusion that inequalities (46)–(49) are, respectively, equivalent to
(50)
α
1
<
1
-
log
⁡
N
A
H
(
a
,
b
)
-
log
⁡
A
(
a
,
b
)
log
⁡
H
(
a
,
b
)
-
log
⁡
A
(
a
,
b
)
=
1
-
log
⁡
[
(
p
+
sinh
⁡
⁡
(
p
)
cosh
⁡
⁡
(
p
)
)
/
(
2
sinh
⁡
⁡
(
p
)
cosh
⁡
⁡
(
p
)
)
]
log
⁡
(
1
/
cosh
⁡
⁡
(
p
)
)
=
1
-
φ
3
(
p
)
<
β
1
,
(51)
α
2
<
1
-
log
⁡
N
H
A
(
a
,
b
)
-
log
⁡
A
(
a
,
b
)
log
⁡
H
(
a
,
b
)
-
log
⁡
A
(
a
,
b
)
=
1
-
log
⁡
[
(
q
+
sin
(
q
)
cos
⁡
(
q
)
)
/
(
2
sin
(
q
)
)
]
log
⁡
[
cos
⁡
(
q
)
]
=
1
-
φ
4
(
q
)
<
β
2
,
(52)
α
3
<
log
⁡
N
C
A
(
a
,
b
)
-
log
⁡
A
(
a
,
b
)
log
⁡
C
(
a
,
b
)
-
log
⁡
A
(
a
,
b
)
=
log
⁡
(
(
r
+
sinh
⁡
(
r
)
cosh
⁡
(
r
)
)
/
(
2
sinh
⁡
(
r
)
)
)
log
⁡
[
cosh
⁡
(
r
)
]
=
φ
5
(
r
)
<
β
3
,
(53)
α
4
<
log
⁡
N
A
C
(
a
,
b
)
-
log
⁡
A
(
a
,
b
)
log
⁡
C
(
a
,
b
)
-
log
⁡
A
(
a
,
b
)
=
log
⁡
[
(
s
+
sin
⁡
(
s
)
cos
⁡
⁡
(
s
)
)
/
(
2
sin
⁡
(
s
)
cos
⁡
⁡
(
s
)
)
]
log
⁡
⁡
[
sec
⁡
(
s
)
]
=
φ
6
(
s
)
<
β
4
.
Therefore, Theorem9 follows easily from (50)–(53) and Lemmas 5–8.Theorem 10.
Letα
5
,
β
5
,
α
6
,
β
6
∈
[
0,1
/
2
]. Then the double inequalities,
(54)
H
[
α
5
a
+
(
1
-
α
5
)
b
,
α
5
b
+
(
1
-
α
5
)
a
]
<
N
A
H
(
a
,
b
)
<
H
[
β
5
a
+
(
1
-
β
5
)
b
,
β
5
b
+
(
1
-
β
5
)
a
]
,
H
[
α
6
a
+
(
1
-
α
6
)
b
,
α
6
b
+
(
1
-
α
6
)
a
]
<
N
H
A
(
a
,
b
)
<
H
[
β
6
a
+
(
1
-
β
6
)
b
,
β
6
b
+
(
1
-
β
6
)
a
]
,
hold for all a
,
b
>
0 with a
≠
b if and only if α
5
≤
1
/
2
-
6
/
6
=
0.09175
…, β
5
≥
1
/
2
-
2
/
4
=
0.1464
…, α
6
≤
1
/
2
-
3
/
6
=
0.2113
…, and β
6
≥
1
/
2
-
1
-
π
/
4
/
2
=
0.2683
….Proof.
Without loss of generality, we assume thata
>
b
>
0. Let λ
∈
[
0,1
/
2
], v
=
(
a
-
b
)
/
(
a
+
b
)
∈
(
0,1
), p
∈
(
0
,
∞
), and q
∈
(
0
,
π
/
2
) be the parameters such that 1
/
cosh
⁡
(
p
)
=
cos
⁡
(
q
)
=
1
-
v
2. Then from (9) and (10) we have
(55)
H
[
λ
a
+
(
1
-
λ
)
b
,
λ
b
+
(
1
-
λ
)
a
]
-
N
A
H
(
a
,
b
)
=
A
(
a
,
b
)
[
1
-
(
1
-
2
λ
)
2
v
2
]
-
1
2
A
(
a
,
b
)
[
1
+
p
sinh
⁡
(
p
)
cosh
⁡
(
p
)
]
=
A
(
a
,
b
)
2
×
[
1
-
2
(
1
-
2
λ
)
2
(
cosh
⁡
⁡
(
p
)
-
1
)
cosh
⁡
⁡
(
p
)
-
p
sinh
⁡
⁡
(
p
)
cosh
⁡
⁡
(
p
)
]
=
A
(
a
,
b
)
[
cosh
⁡
⁡
(
p
)
-
1
]
2
cosh
⁡
⁡
(
p
)
×
[
1
+
sinh
⁡
(
p
)
-
p
sinh
⁡
(
p
)
(
cosh
⁡
(
p
)
-
1
)
-
2
(
1
-
2
λ
)
2
]
=
A
(
a
,
b
)
[
cosh
⁡
(
p
)
-
1
]
2
cosh
⁡
(
p
)
[
1
+
φ
1
(
p
)
-
2
(
1
-
2
λ
)
2
]
,
(56)
H
[
λ
a
+
(
1
-
λ
)
b
,
λ
b
+
(
1
-
λ
)
a
]
-
N
H
A
(
a
,
b
)
=
A
(
a
,
b
)
[
1
-
(
1
-
2
λ
)
2
v
2
]
-
1
2
A
(
a
,
b
)
[
cos
⁡
(
q
)
+
q
sin
(
q
)
]
=
1
2
A
(
a
,
b
)
×
[
2
-
2
(
1
-
2
λ
)
2
(
1
-
cos
⁡
(
q
)
)
-
cos
⁡
(
q
)
-
q
sin
(
q
)
]
=
1
-
cos
⁡
(
q
)
2
A
(
a
,
b
)
[
1
-
2
(
1
-
2
λ
)
2
-
φ
2
(
q
)
]
.
Therefore, Theorem10 follows easily from (55) and (56) together with Lemmas 3 and 4.Theorem 11.
Letα
7
,
β
7
,
α
8
,
β
8
∈
[
1
/
2,1
]. Then the double inequalities,
(57)
C
[
α
7
a
+
(
1
-
α
7
)
b
,
α
7
b
+
(
1
-
α
7
)
a
]
<
N
C
A
(
a
,
b
)
<
C
[
β
7
a
+
(
1
-
β
7
)
b
,
β
7
b
+
(
1
-
β
7
)
a
]
,
C
[
α
8
a
+
(
1
-
α
8
)
b
,
α
8
b
+
(
1
-
α
8
)
a
]
<
N
A
C
(
a
,
b
)
<
H
[
β
8
a
+
(
1
-
β
8
)
b
,
β
8
b
+
(
1
-
β
8
)
a
]
,
hold for all a
,
b
>
0 with a
≠
b if and only if α
7
≤
1
/
2
+
3
/
6
=
0.7886
…, β
7
≥
1
/
2
+
6
3
log
⁡
(
2
+
3
)
/
12
=
0.8082
…, α
8
≤
1
/
2
+
6
/
6
=
0.9082
…, and β
6
≥
1
/
2
+
8
3
π
-
18
/
12
=
0.9210
….Proof.
Without loss of generality, we assume thata
>
b
>
0. Let μ
∈
[
1
/
2,1
], v
=
(
a
-
b
)
/
(
a
+
b
)
∈
(
0,1
), r
∈
(
0
,
log
⁡
(
2
+
3
)
), and s
∈
(
0
,
π
/
3
) be the parameters such that cosh
⁡
(
r
)
=
sec
(
s
)
=
1
+
v
2. Then from (11) and (12) one has
(58)
C
[
μ
a
+
(
1
-
μ
)
b
,
μ
b
+
(
1
-
μ
)
a
]
-
N
C
A
(
a
,
b
)
=
A
(
a
,
b
)
[
1
+
(
2
μ
-
1
)
2
v
2
]
-
1
2
A
(
a
,
b
)
[
cosh
⁡
(
r
)
+
r
sinh
⁡
(
r
)
]
=
1
2
A
(
a
,
b
)
×
[
2
+
2
(
2
μ
-
1
)
2
(
cosh
⁡
⁡
(
r
)
-
1
)
-
cosh
⁡
⁡
(
r
)
-
r
sinh
⁡
⁡
(
r
)
]
=
1
2
A
(
a
,
b
)
[
cosh
⁡
(
r
)
-
1
]
[
2
(
2
μ
-
1
)
2
-
1
+
φ
1
(
r
)
]
,
C
[
μ
a
+
(
1
-
μ
)
b
,
μ
b
+
(
1
-
μ
)
a
]
-
N
A
C
(
a
,
b
)
=
A
(
a
,
b
)
[
1
+
(
2
μ
-
1
)
2
v
2
]
-
1
2
A
(
a
,
b
)
[
1
+
s
sin
(
s
)
cos
⁡
(
s
)
]
=
1
2
A
(
a
,
b
)
×
[
1
+
2
(
2
μ
-
1
)
2
(
sec
⁡
(
s
)
-
1
)
-
s
sin
⁡
(
s
)
cos
⁡
⁡
(
s
)
]
=
1
2
A
(
a
,
b
)
[
sec
(
s
)
-
1
]
[
2
(
2
μ
-
1
)
2
-
1
-
φ
2
(
s
)
]
,
where the functions φ
1 and φ
2 are defined as in Lemmas 3 and 4, respectively.
Note that(59)
φ
1
[
log
⁡
(
2
+
3
)
]
=
1
-
3
log
⁡
⁡
(
2
+
3
)
3
,
φ
2
(
π
3
)
=
4
3
π
-
18
9
.
Therefore, Theorem11 follows easily from Lemmas 3 and 4 together with (58)-(59).Theorem 12.
LetS
A
H
(
a
,
b
)
=
S
[
A
(
a
,
b
)
,
H
(
a
,
b
)
], S
H
A
(
a
,
b
)
=
S
[
H
(
a
,
b
)
,
A
(
a
,
b
)
], S
A
C
(
a
,
b
)
=
S
[
A
(
a
,
b
)
,
C
(
a
,
b
)
], and S
C
A
(
a
,
b
)
=
S
[
C
(
a
,
b
)
,
A
(
a
,
b
)
]. Then the inequalities
(60)
H
(
a
,
b
)
<
S
A
H
(
a
,
b
)
<
N
A
H
(
a
,
b
)
<
S
H
A
(
a
,
b
)
<
N
H
A
(
a
,
b
)
<
A
(
a
,
b
)
<
M
(
a
,
b
)
<
T
(
a
,
b
)
<
S
C
A
(
a
,
b
)
<
N
C
A
(
a
,
b
)
<
Q
(
a
,
b
)
<
S
A
C
(
a
,
b
)
<
N
A
C
(
a
,
b
)
<
C
(
a
,
b
)
hold for all a
,
b
>
0 with a
≠
b.Proof.
It follows from [1, (
3.10
)] and [11, (
4.1
)] together with [9, (
5
), Theorems 2 and 5] that
(61)
N
(
a
,
b
)
>
a
+
2
b
3
>
SB
(
a
,
b
)
,
(62)
H
(
a
,
b
)
<
S
A
H
(
a
,
b
)
<
L
(
a
,
b
)
<
S
H
A
(
a
,
b
)
<
P
(
a
,
b
)
<
A
(
a
,
b
)
<
M
(
a
,
b
)
<
T
(
a
,
b
)
<
S
C
A
(
a
,
b
)
<
Q
(
a
,
b
)
<
S
A
C
(
a
,
b
)
<
C
(
a
,
b
)
.
Therefore, the second, fourth, ninth, and twelfth inequalities follow from (61) and the first, sixth, seventh, eighth, and eleventh inequalities follow from (62) immediately, while the fifth and thirteenth inequalities can be derived from H
(
a
,
b
)
<
A
(
a
,
b
)
<
C
(
a
,
b
) and the fact that N
A
H
(
a
,
b
) and N
A
C
(
a
,
b
) are, respectively, the mean values of A
(
a
,
b
), H
(
a
,
b
), and A
(
a
,
b
), C
(
a
,
b
).
Next, we prove the third and tenth inequalities. In fact, the third inequality can be derived from the following inequalities (63) [14, Theorem 1.2] and (64) [9, Theorem 3] together with A
(
a
,
b
)
>
H
(
a
,
b
). Consider the following:
(63)
N
A
H
(
a
,
b
)
<
1
2
A
(
a
,
b
)
+
1
2
H
(
a
,
b
)
,
(64)
S
H
A
(
a
,
b
)
>
2
π
A
(
a
,
b
)
+
(
1
-
2
π
)
H
(
a
,
b
)
,
while the tenth inequality can be derived from the inequalityN
C
A
(
a
,
b
)
<
C
β
3
(
a
,
b
)
A
1
-
β
3
(
a
,
b
) in Theorem 9 and C
(
a
,
b
)
>
A
(
a
,
b
) together with Q
(
a
,
b
)
=
A
1
/
2
(
a
,
b
)
C
1
/
2
(
a
,
b
), where β
3
=
[
2
log
⁡
(
2
3
+
log
⁡
(
2
+
3
)
)
-
log
⁡
3
]
/
(
2
log
⁡
2
)
-
1
=
0.4648
….Remark 13.
He et al. [14, Theorem 1.2], Xia and Chu [17, Theorem 3.1], and Chu et al. [18, Theorem 2.1] proved that the double inequalities
(65)
α
1
A
(
a
,
b
)
+
(
1
-
α
1
)
H
(
a
,
b
)
<
N
A
H
(
a
,
b
)
<
β
1
A
(
a
,
b
)
+
(
1
-
β
1
)
H
(
a
,
b
)
,
α
2
A
(
a
,
b
)
+
(
1
-
α
2
)
H
(
a
,
b
)
<
N
H
A
(
a
,
b
)
<
β
2
A
(
a
,
b
)
+
(
1
-
β
2
)
H
(
a
,
b
)
,
α
3
A
(
a
,
b
)
+
(
1
-
α
3
)
H
(
a
,
b
)
<
L
(
a
,
b
)
<
β
3
A
(
a
,
b
)
+
(
1
-
β
3
)
H
(
a
,
b
)
,
α
4
A
(
a
,
b
)
+
(
1
-
α
4
)
H
(
a
,
b
)
<
P
(
a
,
b
)
<
β
4
A
(
a
,
b
)
+
(
1
-
β
4
)
H
(
a
,
b
)
hold for all a
,
b
>
0 with a
≠
b if and only if α
1
≤
1
/
3, β
1
≥
1
/
2, α
2
≤
2
/
3, β
2
≥
π
/
4
=
0.7853
…, α
3
≤
0, β
3
≥
2
/
3, α
4
≤
2
/
π
=
0.6366
…, and β
4
≥
5
/
6.
From the above results we clearly see that the mean valuesN
A
H
(
a
,
b
) and L
(
a
,
b
) and N
H
A
(
a
,
b
) and P
(
a
,
b
) are not comparable with each other.



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*Source: 914242-2014-07-23.xml*