# A Sharp Lower Bound for Toader-Qi Mean with Applications

Authors: Zhen-Hang Yang; Yu-Ming Chu
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

We prove that the inequalityT
Q
(
a
,
b
)
>
L
p
(
a
,
b
) holds for all a
,
b
>
0 with a
≠
b if and only if p
≤
3
/
2, where T
Q
(
a
,
b
)
=
2
/
π
∫
0
π
/
2
a
c
o
s
2
θ
b
s
i
n
2
θ
d
θ, L
p
(
a
,
b
)
=
[
(
b
p
-
a
p
)
/
(
p
(
b
-
a
)
)
]
1
/
p
(
p
≠
0
), and L
0
(
a
,
b
)
=
a
b are, respectively, the Toader-Qi and p-order logarithmic means of a and b. As applications, we find two fine inequalities chains for certain bivariate means.


---

## Body

## 1. Introduction

Letp
∈
R and a
,
b
>
0 with a
≠
b. Then the Toader-Qi mean T
Q
(
a
,
b
) [1–3] and p-order logarithmic mean L
p
(
a
,
b
) are defined by(1)
T
Q
a
,
b
=
2
π
∫
0
π
/
2
a
c
o
s
2
θ
b
s
i
n
2
θ
d
θ
,
(2)
L
p
a
,
b
=
b
p
-
a
p
p
log
⁡
b
-
log
⁡
a
1
/
p
p
≠
0
,
L
0
a
,
b
=
l
i
m
p
→
0
L
p
a
,
b
=
a
b
,respectively. In particular, L
1
(
a
,
b
)
=
L
(
a
,
b
) is the classical logarithmic mean of a and b.It is well-known that thep-order logarithmic mean L
p
(
a
,
b
) is continuous and strictly increasing with respect to p
∈
R for fixed a
,
b
>
0 with a
≠
b. Recently, the Toader-Qi and p-order logarithmic means have been the subject of intensive research. In particular, many remarkable inequalities for the Toader-Qi and p-order logarithmic means can be found in the literature [2–7].In [2], Qi et al. proved that the identity(3)
T
Q
a
,
b
=
a
b
I
0
1
2
log
⁡
b
aand the inequalities(4)
L
a
,
b
<
T
Q
a
,
b
<
A
a
,
b
+
G
a
,
b
2
<
2
A
a
,
b
+
G
a
,
b
3
<
I
a
,
bhold for all a
,
b
>
0 with a
≠
b, where(5)
I
0
t
=
∑
n
=
0
∞
t
2
n
2
2
n
n
!
2is the modified Bessel function of the first kind [8] and A
(
a
,
b
)
=
(
a
+
b
)
/
2, G
(
a
,
b
)
=
a
b and I
(
a
,
b
)
=
(
b
b
/
a
a
)
1
/
(
b
-
a
)
/
e are, respectively, the classical arithmetic, geometric, and identric means of a and b.In [3], Yang proved that the double inequalities (6)
2
A
a
,
b
L
a
,
b
π
<
T
Q
a
,
b
<
A
a
,
b
L
a
,
b
,
(7)
A
1
/
4
a
,
b
L
3
/
4
a
,
b
<
T
Q
a
,
b
<
1
4
A
a
,
b
+
3
4
L
a
,
band conjectured that the inequalities(8)
T
Q
a
,
b
<
I
1
/
2
a
,
b
L
1
/
2
a
,
b
,
(9)
T
Q
a
,
b
>
L
3
/
2
a
,
bhold for all a
,
b
>
0 with a
≠
b. Inequality (8) was proved by Yang et al. in [9].Letb
>
a
>
0 and t
=
(
log
⁡
b
-
log
⁡
a
)
/
2
>
0. Then from (1)–(3) we clearly see that(10)
L
p
a
,
b
=
a
b
sinh
⁡
p
t
p
t
1
/
p
p
≠
0
,
T
Q
a
,
b
=
2
a
b
π
∫
0
π
/
2
e
t
c
o
s
2
θ
d
θ
=
a
b
I
0
t
=
2
a
b
π
∫
0
π
/
2
cosh
⁡
t
cos
⁡
θ
d
θ
=
2
a
b
π
∫
0
π
/
2
cosh
⁡
t
sin
⁡
θ
d
θ
.The main purpose of this paper is to give a positive answer to the conjecture given by (9). As applications, we present two fine inequalities chains for certain bivariate means and a lower bound for the kernel function of the Szász-Mirakjan-Durrmeyer operator.


## 2. Lemmas

In order to prove our main result we need several lemmas, which we present in this section.Lemma 1 (see [10]).
The double inequality(11)
1
x
+
a
1
-
a
<
Γ
x
+
a
Γ
x
+
1
<
1
x
1
-
aholds for all x
>
0 and a
∈
(
0,1
), where Γ
(
x
)
=
∫
0
∞
‍
e
-
t
t
x
-
1
d
t is the classical Euler gamma function.Lemma 2 (see [3]).
LetI
0
(
t
) be defined by (5). Then the identity (12)
I
0
2
t
=
∑
n
=
0
∞
2
n
!
2
2
n
n
!
4
t
2
nholds for all t
∈
R.Lemma 3 (see [3]).
The Wallis ratio(13)
W
n
=
2
n
-
1
!
!
2
n
!
!
=
2
n
!
2
2
n
n
!
2
=
Γ
n
+
1
/
2
Γ
1
/
2
Γ
n
+
1is strictly decreasing and log-convex with respect to all integers n
≥
0.Lemma 4.
The identity(14)
∑
k
=
0
n
a
2
k
2
k
!
2
n
-
2
k
!
=
a
+
1
2
n
+
a
-
1
2
n
2
2
n
!holds for all a
∈
R and n
∈
N.Proof.
Letn
k
=
n
!
/
k
!
(
n
-
k
)
! be the number of combinations of n objects taken k at a time. Then from the well-known binomial theorem we have(15)
a
+
1
2
n
=
∑
k
=
0
2
n
2
n
k
a
k
=
∑
k
=
0
n
2
n
2
k
a
2
k
+
∑
k
=
1
n
2
n
2
k
-
1
a
2
k
-
1
,
a
-
1
2
n
=
∑
k
=
0
2
n
2
n
k
-
1
2
n
-
k
a
k
=
∑
k
=
0
n
2
n
2
k
a
2
k
-
∑
k
=
1
n
2
n
2
k
-
1
a
2
k
-
1
.
Equation (15) leads to (16)
a
+
1
2
n
+
a
-
1
2
n
2
=
∑
k
=
0
n
2
n
2
k
a
2
k
=
∑
k
=
0
n
2
n
!
a
2
k
2
k
!
2
n
-
2
k
!
.Lemma 5.
Letk
,
n
∈
N with k
≤
n and(17)
u
k
,
n
=
2
k
!
2
2
n
k
!
4
n
-
k
!
2
.Then(18)
u
k
,
n
>
2
2
π
π
n
+
1
2
n
+
1
2
2
k
2
k
!
2
n
-
2
k
!for all n
≥
8.Proof.
LetW
n be defined by (13). Then it follows from Lemmas 1 and 3 together with (17) and Γ
(
1
/
2
)
=
π that (19)
u
k
,
n
=
W
k
2
W
n
-
k
2
2
k
2
k
!
2
n
-
2
k
!
≥
W
k
W
n
/
2
2
2
2
k
2
k
!
2
n
-
2
k
!
≥
W
n
W
n
/
2
2
2
2
k
2
k
!
2
n
-
2
k
!
=
1
π
π
Γ
n
+
1
/
2
Γ
n
+
1
Γ
n
/
2
+
1
/
2
Γ
n
/
2
+
1
2
2
2
k
2
k
!
2
n
-
2
k
!
>
1
π
π
1
n
+
1
/
2
1
n
/
2
+
1
/
2
2
2
2
k
2
k
!
2
n
-
2
k
!
=
2
2
π
π
n
+
1
2
n
+
1
2
2
k
2
k
!
2
n
-
2
k
!for all n
≥
8 and 0
≤
k
≤
n.


## 3. Main Result

Theorem 6.
The inequality(20)
T
Q
a
,
b
>
L
p
a
,
bholds for all a
,
b
>
0 with a
≠
b if and only if p
≤
3
/
2.Proof.
Since both the Toader-Qi meanT
Q
(
a
,
b
) and p-order logarithmic mean L
p
(
a
,
b
) are symmetric and homogeneous and T
Q
(
a
,
b
)
>
L
(
a
,
b
) and L
p
(
a
,
b
) is strictly increasing with respect to p
∈
R for all a
,
b
>
0 with a
≠
b, without loss of generality, we assume that p
>
1 and b
>
a
>
0. Let t
=
(
log
⁡
b
-
log
⁡
a
)
/
2
>
0. Then it follows from (10) that inequality (20) is equivalent to(21)
I
0
t
>
sinh
⁡
p
t
p
t
1
/
pfor all t
>
0.
If inequality (21) holds for all t
>
0. Then (5) and (21) lead to (22)
l
i
m
t
→
0
+
I
0
t
-
sinh
⁡
p
t
/
p
t
1
/
p
t
2
=
-
1
6
p
-
3
2
≥
0
,which gives p
≤
3
/
2.
Next, we only need to prove that inequality (21) holds for p
=
3
/
2 and all t
>
0; that is(23)
I
0
3
t
>
sinh
⁡
3
t
/
2
3
t
/
2
2
.
It follows from (5) and Lemma 2 that(24)
I
0
3
t
=
∑
n
=
0
∞
t
2
n
2
2
n
n
!
2
3
=
∑
n
=
0
∞
2
n
!
t
2
n
2
2
n
n
!
4
∑
n
=
0
∞
t
2
n
2
2
n
n
!
2
=
∑
n
=
0
∞
∑
k
=
0
n
2
k
!
2
2
k
k
!
4
1
2
2
n
-
k
n
-
k
!
2
t
2
n
=
∑
n
=
0
∞
∑
k
=
0
n
u
k
,
n
t
2
n
,where u
k
,
n is defined as in (17).
Note that(25)
sinh
⁡
3
t
/
2
3
t
/
2
2
=
2
cosh
⁡
3
t
-
1
9
t
2
=
2
∑
n
=
0
∞
3
2
n
t
2
n
2
n
+
2
!
.
Let(26)
v
n
=
∑
k
=
0
n
u
k
,
n
-
2
×
3
2
n
2
n
+
2
!
.
Then simple computations lead to(27)
v
0
=
v
1
=
0
,
v
2
=
3
320
,
v
3
=
113
26880
,
v
4
=
2057
2867200
,
v
5
=
1741
25231360
,
v
6
=
4335377
991895224320
,
v
7
=
2186227
11021058048000
.
From Lemmas4 and 5 together with (24)–(26), we have(28)
I
0
3
t
-
sinh
⁡
3
t
/
2
3
t
/
2
2
=
∑
n
=
0
∞
v
n
t
2
n
,
v
n
>
2
2
π
π
n
+
1
2
n
+
1
3
2
n
+
1
2
2
n
!
-
2
×
3
2
n
2
n
+
2
!
=
2
2
n
+
1
-
π
π
3
2
n
+
2
2
n
+
1
π
π
n
+
1
2
n
+
1
!
>
0for all n
≥
8.
Therefore, inequality (23) follows from (27) and (28).Remark 7.
Theorem6 gives a positive answer to the conjecture given by (9).Remark 8.
It follows from (23) that the inequality (29)
I
0
3
t
>
2
cosh
⁡
3
t
-
1
9
t
2holds for all t
≠
0.


## 4. Applications

Fora
,
b
>
0, the Toader mean T
(
a
,
b
) [1] and arithmetic-geometric mean A
G
M
(
a
,
b
) [11] are, respectively, defined by (30)
T
a
,
b
=
2
π
∫
0
π
/
2
a
2
cos
2
⁡
θ
+
b
2
sin
2
⁡
θ
d
θ
,
A
G
M
a
,
b
=
l
i
m
n
→
∞
a
n
=
l
i
m
n
→
∞
b
n
,where a
n and b
n are given by (31)
a
0
=
a
,
b
0
=
b
,
a
n
+
1
=
a
n
+
b
n
2
=
A
a
n
,
b
n
,
b
n
+
1
=
a
n
b
n
=
G
a
n
,
b
n
.LetT
p
(
a
,
b
)
=
T
1
/
p
a
p
,
b
p and I
q
(
a
,
b
)
=
I
1
/
q
a
q
,
b
q be the p-order Toader and q-order identric means of a and b, respectively. Then Theorem 6 leads to two fine inequalities chains for certain bivariate means.Theorem 9.
The inequalities(32)
L
a
,
b
<
A
G
M
a
,
b
<
A
1
/
4
a
,
b
L
3
/
4
a
,
b
<
L
3
/
2
a
,
b
<
T
Q
a
,
b
<
1
4
A
a
,
b
+
3
4
L
a
,
b
<
1
2
L
a
,
b
+
1
2
I
a
,
b
<
1
2
A
a
,
b
+
1
2
G
a
,
b
<
T
1
/
3
a
,
b
<
I
3
/
4
a
,
b
,
L
a
,
b
<
A
G
M
a
,
b
<
A
1
/
4
a
,
b
L
3
/
4
a
,
b
<
L
3
/
2
a
,
b
<
T
Q
a
,
b
<
L
1
/
2
a
,
b
I
1
/
2
a
,
b
<
1
2
L
a
,
b
+
1
2
I
a
,
b
<
1
2
A
a
,
b
+
1
2
G
a
,
b
<
T
1
/
3
a
,
b
<
I
3
/
4
a
,
bhold for all a
,
b
>
0 with a
≠
b.Proof.
The following inequalities can be found in the literature [3, 4, 7, 12–14]:(33)
A
a
,
b
+
G
a
,
b
2
<
T
1
/
3
a
,
b
<
I
3
/
4
a
,
b
,
(34)
L
a
,
b
<
A
G
M
a
,
b
<
L
3
/
4
a
,
b
A
1
/
4
a
,
b
<
L
3
/
2
a
,
b
,
(35)
I
a
,
b
>
L
a
,
b
+
A
a
,
b
2
,
L
a
,
b
+
I
a
,
b
<
A
a
,
b
+
G
a
,
bfor all a
,
b
>
0 with a
≠
b.
It follows from (35) that(36)
A
a
,
b
+
G
a
,
b
2
>
I
a
,
b
+
L
a
,
b
2
>
3
4
L
a
,
b
+
1
4
A
a
,
bfor all a
,
b
>
0 with a
≠
b.
Therefore, inequality (32) follows easily from (7), (8), (33), (34), (36), and Theorem 6.Remark 10.
Letb
>
a
>
0 and t
=
(
log
⁡
b
-
log
⁡
a
)
/
2
>
0. Then simple computations lead to(37)
L
a
,
b
a
b
=
sinh
⁡
t
t
,
I
a
,
b
a
b
=
e
t
cosh
⁡
t
/
sinh
⁡
t
-
1
,
A
a
,
b
a
b
=
cosh
⁡
t
.
Note that(38)
lim
t
→
0
+
⁡
sinh
⁡
t
/
t
e
t
cosh
⁡
t
/
sinh
⁡
t
-
1
-
3
sinh
⁡
t
/
4
t
+
cosh
⁡
t
/
4
2
t
4
=
1
720
,
lim
t
→
∞
⁡
sinh
⁡
t
t
e
t
cosh
⁡
t
/
sinh
⁡
t
-
1
-
3
sinh
⁡
t
4
t
+
cosh
⁡
t
4
2
=
-
∞
.
Inequalities (37) and (38) imply that there exist small enough δ
>
0 and large enough M
>
1 such that (39)
I
1
/
2
a
,
b
L
1
/
2
a
,
b
>
1
4
A
a
,
b
+
3
4
L
a
,
bfor all b
>
a
>
0 with b
/
a
∈
(
1,1
+
δ
) and (40)
I
1
/
2
a
,
b
L
1
/
2
a
,
b
<
1
4
A
a
,
b
+
3
4
L
a
,
bfor all b
>
a
>
0 with b
/
a
∈
(
M
,
∞
).
Letx
∈
[
0
,
∞
), n
>
0, k
≥
0, p
n
,
k
(
x
)
=
(
n
x
)
k
e
-
n
x
/
k
!, and f
∈
L
p
(
[
0
,
∞
)
)
(
1
≤
p
≤
∞
). Then the kernel function T
n
(
x
,
y
) of the Szász-Mirakjan-Durrmeyer operator [15] (41)
M
n
f
;
x
=
∑
k
=
0
∞
f
,
p
n
,
k
1
,
p
n
,
k
p
n
,
k
x
=
n
f
,
T
n
x
,
·
∫
0
∞
T
n
x
,
y
f
y
d
yis given by(42)
T
n
x
,
y
=
∑
k
=
0
∞
p
n
,
k
x
p
n
,
k
y
=
e
-
n
x
+
y
I
0
2
n
x
y
.
Berdysheva [16] proved that T
n
(
x
,
y
) is completely monotonic with respect to n
>
0 for fixed x
,
y
∈
[
0
,
∞
) and (43)
T
n
x
,
y
≤
e
-
n
x
-
y
2for all x
,
y
∈
[
0
,
∞
).From Remark8 and (42), we get a lower bound for the kernel function T
n
(
x
,
y
) immediately.Corollary 11.
The inequality(44)
T
n
x
,
y
>
e
-
n
x
+
y
cosh
⁡
6
n
x
y
-
1
18
n
2
x
y
1
/
3holds for all x
,
y
∈
(
0
,
∞
).



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*Source: 4165601-2016-01-17.xml*