# A New Extension of Hardy-Hilbert’s Inequality in the Whole Plane

Authors: Bicheng Yang; Qiang Chen
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

By the use of weight coefficients and Hermite-Hadamard’s inequality, a new extension of Hardy-Hilbert’s inequality in the whole plane with multiparameters and a best possible constant factor is given. The equivalent forms, the operator expressions, and a few particular inequalities are considered.


---

## Body

## 1. Introduction

Suppose thatp
>
1, 1
/
p
+
1
/
q
=
1, a
m
,
b
n
≥
0, a
=
a
m
m
=
1
∞
∈
l
p, b
=
b
n
n
=
1
∞
∈
l
q, a
p
=
∑
m
=
1
∞
a
m
p
1
/
p
>
0, and b
q
>
0
. We have the following well known Hardy-Hilbert’s inequality:(1)
∑
n
=
1
∞
∑
m
=
1
∞
a
m
b
n
m
+
n
<
π
sin
⁡
π
/
p
a
p
b
q
,where the constant factor π
/
sin
⁡
(
π
/
p
) is the best possible one (cf. [1]). The more accurate form of (1) was given as follows (cf. [2], Theorem 323):(2)
∑
n
=
1
∞
∑
m
=
1
∞
a
m
b
n
m
+
n
-
2
α
<
π
s
i
n
π
/
p
a
p
b
q
0
≤
α
≤
1
2
,where the constant factor π
/
s
i
n
(
π
/
p
) is still the best possible one. For α
=
0, inequality (2) reduces to (1).In 2011, Yang gave an extension of (2) as follows (cf. [3]): If 0
<
λ
1, λ
2
≤
1, λ
1
+
λ
2
=
λ, a
m
,
b
n
≥
0, a
p
,
φ
=
∑
m
=
1
∞
m
-
α
p
1
-
λ
1
-
1
a
m
p
1
/
p
∈
(
0
,
∞
), and b
q
,
ψ
=
∑
n
=
1
∞
n
-
α
q
1
-
λ
2
-
1
b
n
q
1
/
q
∈
(
0
,
∞
), then(3)
∑
n
=
1
∞
∑
m
=
1
∞
a
m
b
n
m
+
n
-
2
α
λ
<
B
λ
1
,
λ
2
a
p
,
φ
b
q
,
ψ
0
≤
α
≤
1
2
,where the constant factor B
(
λ
1
,
λ
2
) is the best possible one and B
(
u
,
v
) is the beta function defined by (cf. [4])(4)
B
u
,
v
≔
∫
0
∞
1
1
+
t
u
+
v
t
u
-
1
d
t
u
,
v
>
0
.For λ
=
1, λ
1
=
1
/
q, and λ
2
=
1
/
p, (3) reduces to (2). Some other results related to (1)–(3) are provided by [5–22].In this paper, by the use of weight coefficients and Hermite-Hadamard’s inequality, an extension of (3) in the whole plane is given as follows: For ξ
,
η
∈
0
,
1
/
2, a
m
,
b
n
≥
0, ∑
m
=
1
∞
m
-
ξ
p
(
1
-
λ
1
)
-
1
a
m
p
∈
(
0
,
∞
), and ∑
n
=
1
∞
n
-
η
q
(
1
-
λ
2
)
-
1
b
n
q
∈
(
0
,
∞
), we have(5)
∑
n
=
1
∞
∑
m
=
1
∞
1
m
-
ξ
+
n
-
η
λ
a
m
b
n
<
2
B
λ
1
,
λ
2
∑
m
=
1
∞
m
-
ξ
p
1
-
λ
1
-
1
a
m
p
1
/
p
∑
n
=
1
∞
n
-
η
q
1
-
λ
2
-
1
b
n
q
1
/
q
.Moreover, a generation of (5) with multiparameters and a best possible constant factor is proved. The equivalent forms, the operator expressions, and a few particular inequalities are also considered.


## 2. Some Lemmas

First, we make appointment thatN
=
{
1,2
,
…
}, p
>
1, 1
/
p
+
1
/
q
=
1, α
,
β
∈
(
0
,
π
), ξ
,
η
∈
0
,
1
/
2, λ
1
,
λ
2
>
0, λ
1
+
λ
2
=
λ, and, for x
,
y
>
1
/
2, (6)
k
x
,
y
≔
1
x
-
ξ
+
x
-
ξ
cos
⁡
α
+
y
-
η
+
y
-
η
cos
⁡
β
λ
.In particular, for α
=
β
=
π
/
2, we indicate(7)
h
x
,
y
≔
1
x
-
ξ
+
y
-
η
λ
,
x
,
y
>
1
2
.Definition 1.
Define the following weight coefficients:(8)
ω
λ
2
,
m
≔
∑
n
=
1
∞
k
m
,
n
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
n
-
η
+
n
-
η
cos
⁡
β
1
-
λ
2
,
m
∈
N
,
(9)
ϖ
λ
1
,
n
≔
∑
m
=
1
∞
k
m
,
n
n
-
η
+
n
-
η
cos
⁡
β
λ
2
m
-
ξ
+
m
-
ξ
cos
⁡
α
1
-
λ
1
,
n
∈
N
,where ∑
|
j
|
=
1
∞
⋯
=
∑
j
=
-
1
-
∞
⋯
+
∑
j
=
1
∞
⋯
(
j
=
m
,
n
)
.Lemma 2.
Ifλ
2
≤
1, then, for k
β
(
λ
1
)
≔
2
B
λ
1
,
λ
2
csc
2
⁡
β, one has(10)
k
β
λ
1
1
-
θ
λ
2
,
m
<
ω
λ
2
,
m
<
k
β
λ
1
,
m
∈
N
,where(11)
θ
λ
2
,
m
≔
1
B
λ
1
,
λ
2
∫
0
1
+
η
1
+
cos
⁡
β
/
m
-
ξ
+
m
-
ξ
cos
⁡
α
u
λ
2
-
1
1
+
u
λ
d
u
=
O
1
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
2
∈
0,1
,
m
∈
N
.Proof.
Forx
>
1
/
2, we set(12)
k
1
x
,
y
≔
1
x
-
ξ
+
x
-
ξ
cos
⁡
α
+
y
-
η
cos
⁡
β
-
1
λ
,
y
<
-
1
2
,
k
2
x
,
y
≔
1
x
-
ξ
+
x
-
ξ
cos
⁡
α
+
y
-
η
1
+
cos
⁡
β
λ
,
y
>
1
2
,and then, for y
>
1
/
2,(13)
k
1
x
,
-
y
=
1
x
-
ξ
+
x
-
ξ
cos
⁡
α
+
y
+
η
1
-
cos
⁡
β
λ
.
We find(14)
ω
λ
2
,
m
=
∑
n
=
-
1
-
∞
k
1
m
,
n
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
n
-
η
cos
⁡
β
-
1
1
-
λ
2
+
∑
n
=
1
∞
k
2
m
,
n
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
n
-
η
1
+
cos
⁡
β
1
-
λ
2
=
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
1
-
cos
⁡
β
1
-
λ
2
∑
n
=
1
∞
k
1
m
,
-
n
n
+
η
1
-
λ
2
+
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
1
+
cos
⁡
β
1
-
λ
2
∑
n
=
1
∞
k
2
m
,
n
n
-
η
1
-
λ
2
.
It is evident that, for fixedm
∈
N, λ
2
≤
1
(
λ
>
0
), both k
(
1
)
(
m
,
-
y
)
/
y
+
η
1
-
λ
2 and k
(
2
)
(
m
,
y
)
/
y
-
η
1
-
λ
2 are strictly decreasing and strictly convex with respect to y
∈
(
1
/
2
,
∞
), satisfying(15)
d
d
y
k
i
m
,
-
1
i
y
y
+
-
1
i
η
1
-
λ
2
<
0
,
d
2
d
y
2
k
i
m
,
-
1
i
y
y
+
-
1
i
η
1
-
λ
2
>
0
,
i
=
1,2
.By Hermite-Hadamard’s inequality (cf. [23]), we find(16)
ω
λ
2
,
m
<
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
1
-
cos
⁡
β
1
-
λ
2
∫
1
/
2
∞
k
1
m
,
-
y
y
+
η
1
-
λ
2
d
y
+
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
1
+
cos
⁡
β
1
-
λ
2
∫
1
/
2
∞
k
2
m
,
y
y
-
η
1
-
λ
2
d
y
≤
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
1
-
cos
⁡
β
1
-
λ
2
∫
-
η
∞
k
1
m
,
-
y
y
+
η
1
-
λ
2
d
y
+
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
1
+
cos
⁡
β
1
-
λ
2
∫
η
∞
k
2
m
,
y
y
-
η
1
-
λ
2
d
y
.Setting u
= (
y
+
η
)
(
1
-
cos
⁡
β
)
/
m
-
ξ
+
m
-
ξ
cos
⁡
α (
(
y
-
η
)
(
1
+
cos
⁡
β
)
/
m
-
ξ
+
m
-
ξ
cos
⁡
α
) in the above first (second) integral, by simplification, we find(17)
ω
λ
2
,
m
<
1
1
-
cos
⁡
β
+
1
1
+
cos
⁡
β
∫
0
∞
u
λ
2
-
1
1
+
u
λ
d
u
=
2
B
λ
1
,
λ
2
csc
2
⁡
β
=
k
β
λ
1
.
By (14), since both k
(
1
)
(
m
,
-
y
)
/
y
+
η
1
-
λ
2 and k
(
2
)
(
m
,
y
)
/
y
-
η
1
-
λ
2 are strictly decreasing, we still have(18)
ω
λ
2
,
m
>
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
1
-
cos
⁡
β
1
-
λ
2
∫
1
∞
k
1
m
,
-
y
y
+
η
1
-
λ
2
d
y
+
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
1
+
cos
⁡
β
1
-
λ
2
∫
1
∞
k
2
m
,
y
y
-
η
1
-
λ
2
d
y
≥
1
1
-
cos
⁡
β
∫
1
+
η
1
+
cos
⁡
β
/
m
-
ξ
+
m
-
ξ
cos
⁡
α
∞
u
λ
2
-
1
1
+
u
λ
d
u
+
1
1
+
cos
⁡
β
∫
1
+
η
1
+
cos
⁡
β
/
m
-
ξ
+
m
-
ξ
cos
⁡
α
∞
u
λ
2
-
1
1
+
u
λ
d
u
=
k
β
λ
2
1
-
θ
λ
2
,
m
>
0
.
We obtain(19)
0
<
θ
λ
2
,
m
=
1
B
λ
1
,
λ
2
∫
0
1
+
η
1
+
cos
⁡
β
/
m
-
ξ
+
m
-
ξ
cos
⁡
α
u
λ
2
-
1
1
+
u
λ
d
u
<
1
B
λ
1
,
λ
2
∫
0
1
+
η
1
+
cos
⁡
β
/
m
-
ξ
+
m
-
ξ
cos
⁡
α
u
λ
2
-
1
d
u
=
1
λ
2
B
λ
1
,
λ
2
1
+
η
1
+
cos
⁡
β
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
2
,and then we have (10) and (11).In the same way, we still have the following.Lemma 3.
Ifλ
1
≤
1, then, for k
α
(
λ
1
)
=
2
B
λ
1
,
λ
2
csc
2
⁡
α, one has(20)
k
α
λ
1
1
-
ϑ
λ
1
,
n
<
ϖ
λ
1
,
n
<
k
α
λ
1
,
n
∈
N
,where(21)
ϑ
λ
1
,
n
≔
1
B
λ
1
,
λ
2
∫
0
1
+
ξ
1
+
cos
⁡
α
/
n
-
η
+
n
-
η
cos
⁡
β
u
λ
1
-
1
1
+
u
λ
d
u
=
O
1
n
-
η
+
n
-
η
cos
⁡
β
λ
1
∈
0,1
,
n
∈
N
.Lemma 4.
Ifζ
∈
0
,
1
/
2 and θ
∈
(
0
,
π
), then, for ρ
>
0,(22)
H
ρ
ζ
,
θ
≔
∑
n
=
1
∞
1
n
-
ζ
+
n
-
ζ
cos
⁡
θ
1
+
ρ
=
1
+
o
1
ρ
1
1
+
cos
⁡
θ
1
+
ρ
+
1
1
-
cos
⁡
θ
1
+
ρ
ρ
⟶
0
+
.Proof.
We have(23)
H
ρ
ζ
,
θ
=
∑
n
=
-
1
-
∞
1
n
-
ζ
cos
⁡
θ
-
1
1
+
ρ
+
∑
n
=
1
∞
1
n
-
ζ
cos
⁡
θ
+
1
1
+
ρ
=
1
1
-
cos
⁡
θ
1
+
ρ
∑
n
=
1
∞
1
n
+
ζ
1
+
ρ
+
1
1
+
cos
⁡
θ
1
+
ρ
∑
n
=
1
∞
1
n
-
ζ
1
+
ρ
.For a
=
1
/
1
-
ζ
1
+
ρ, we find(24)
H
ρ
ζ
,
θ
≤
1
1
-
cos
⁡
θ
1
+
ρ
+
1
1
+
cos
⁡
θ
1
+
ρ
a
+
∑
n
=
2
∞
1
n
-
ζ
1
+
ρ
<
1
1
-
cos
⁡
θ
1
+
ρ
+
1
1
+
cos
⁡
θ
1
+
ρ
a
+
∫
1
∞
d
y
y
-
ζ
1
+
ρ
=
1
ρ
1
1
-
cos
⁡
θ
1
+
ρ
+
1
1
+
cos
⁡
θ
1
+
ρ
1
+
a
ρ
+
1
1
-
ζ
ρ
-
1
,
H
ρ
ζ
,
θ
≥
1
1
-
cos
⁡
θ
1
+
ρ
+
1
1
+
cos
⁡
θ
1
+
ρ
∑
n
=
1
∞
1
n
+
ζ
1
+
ρ
>
1
1
-
cos
⁡
θ
1
+
ρ
+
1
1
+
cos
⁡
θ
1
+
ρ
∫
1
∞
d
y
y
+
ζ
1
+
ρ
=
1
+
1
+
ζ
-
ρ
-
1
ρ
1
1
-
cos
⁡
θ
1
+
ρ
+
1
1
+
cos
⁡
θ
1
+
ρ
.Hence, we have (22).


## 3. Main Results and Operation Expressions

Theorem 5.
Ifλ
1
,
λ
2
≤
1, a
m
,
b
n
≥
0
(
m
,
n
∈
N
),(25)
0
<
∑
m
=
1
∞
m
-
ξ
+
m
-
ξ
cos
⁡
α
p
1
-
λ
1
-
1
a
m
p
<
∞
,
0
<
∑
n
=
1
∞
n
-
η
+
n
-
η
cos
⁡
β
q
1
-
λ
2
-
1
b
n
q
<
∞
,
k
λ
1
≔
k
β
1
/
p
λ
1
k
α
1
/
q
λ
1
=
2
B
λ
1
,
λ
2
csc
2
/
p
⁡
β
csc
2
/
q
⁡
α
,then one has the following equivalent inequalities:(26)
I
≔
∑
n
=
1
∞
∑
m
=
1
∞
k
m
,
n
a
m
b
n
<
k
λ
1
∑
m
=
1
∞
m
-
ξ
+
m
-
ξ
cos
⁡
α
p
1
-
λ
1
-
1
a
m
p
1
/
p
∑
n
=
1
∞
n
-
η
+
n
-
η
cos
⁡
β
q
1
-
λ
2
-
1
b
n
q
1
/
q
,
(27)
J
≔
∑
n
=
1
∞
n
-
η
+
n
-
η
cos
⁡
β
p
λ
2
-
1
∑
m
=
1
∞
k
m
,
n
a
m
p
1
/
p
<
k
λ
1
∑
m
=
1
∞
m
-
ξ
+
m
-
ξ
cos
⁡
α
p
1
-
λ
1
-
1
a
m
p
1
/
p
.Proof.
By Hölder’s inequality (cf. [23]) and (9), we have(28)
∑
m
=
1
∞
k
m
,
n
a
m
p
=
∑
m
=
1
∞
k
m
,
n
m
-
ξ
+
m
-
ξ
cos
⁡
α
1
-
λ
1
/
q
n
-
η
+
n
-
η
cos
⁡
β
1
-
λ
2
/
p
a
m
n
-
η
+
n
-
η
cos
⁡
β
1
-
λ
2
/
p
m
-
ξ
+
m
-
ξ
cos
⁡
α
1
-
λ
1
/
q
p
≤
∑
m
=
1
∞
k
m
,
n
m
-
ξ
+
m
-
ξ
cos
⁡
α
1
-
λ
1
p
/
q
n
-
η
+
n
-
η
cos
⁡
β
1
-
λ
2
a
m
p
∑
m
=
1
∞
k
m
,
n
n
-
η
+
n
-
η
cos
⁡
β
1
-
λ
2
q
/
p
m
-
ξ
+
m
-
ξ
cos
⁡
α
1
-
λ
1
p
-
1
=
ϖ
λ
1
,
n
p
-
1
n
-
η
+
n
-
η
cos
⁡
β
p
λ
2
-
1
∑
m
=
1
∞
k
m
,
n
m
-
ξ
+
m
-
ξ
cos
⁡
α
1
-
λ
1
p
/
q
n
-
η
+
n
-
η
cos
⁡
β
1
-
λ
2
a
m
p
.By (20), we have(29)
J
<
k
α
1
/
q
λ
1
∑
n
=
1
∞
∑
m
=
1
∞
k
m
,
n
m
-
ξ
+
m
-
ξ
cos
⁡
α
1
-
λ
1
p
/
q
n
-
η
+
n
-
η
cos
⁡
β
1
-
λ
2
a
m
p
1
/
p
=
k
α
1
/
q
λ
1
∑
m
=
1
∞
∑
n
=
1
∞
k
m
,
n
m
-
ξ
+
m
-
ξ
cos
⁡
α
1
-
λ
1
p
/
q
n
-
η
+
n
-
η
cos
⁡
β
1
-
λ
2
a
m
p
1
/
p
=
k
α
1
/
q
λ
1
∑
m
=
1
∞
ω
λ
2
,
m
m
-
ξ
+
m
-
ξ
cos
⁡
α
p
1
-
λ
1
-
1
a
m
p
1
/
p
.By (10), we have (27).
By Hölder’s inequality (cf. [23]), we have (30)
I
=
∑
n
=
1
∞
n
-
η
+
n
-
η
cos
⁡
β
λ
2
-
1
/
p
∑
m
=
1
∞
k
m
,
n
a
m
n
-
η
+
n
-
η
cos
⁡
β
1
/
p
-
λ
2
b
n
≤
J
∑
n
=
1
∞
n
-
η
+
n
-
η
cos
⁡
β
q
1
-
λ
2
-
1
b
n
q
1
/
q
.Then by (27), we have (26).
On the other hand, assuming that (26) is valid, we set(31)
b
n
≔
n
-
η
+
n
-
η
cos
⁡
β
p
λ
2
-
1
∑
m
=
1
∞
k
m
,
n
a
m
p
-
1
,
n
∈
N
.Then it follows that (32)
J
=
∑
n
=
1
∞
n
-
η
+
n
-
η
cos
⁡
β
q
1
-
λ
2
-
1
b
n
q
1
/
p
.By (29), we find J
<
∞
. If J
=
0, then (27) is evidently valid; if J
>
0, then, by (26), we have (33)
0
<
∑
n
=
1
∞
n
-
η
+
n
-
η
cos
⁡
β
q
1
-
λ
2
-
1
b
n
q
=
J
p
=
I
<
k
λ
1
∑
m
=
1
∞
m
-
ξ
+
m
-
ξ
cos
⁡
α
p
1
-
λ
1
-
1
a
m
p
1
/
p
∑
n
=
1
∞
n
-
η
+
n
-
η
cos
⁡
β
q
1
-
λ
2
-
1
b
n
q
1
/
q
,
J
=
∑
n
=
1
∞
n
-
η
+
n
-
η
cos
⁡
β
q
1
-
λ
2
-
1
b
n
q
1
/
p
<
k
λ
1
∑
m
=
1
∞
m
-
ξ
+
m
-
ξ
cos
⁡
α
p
1
-
λ
1
-
1
a
m
p
1
/
p
;namely, (27) follows, which is equivalent to (26).Theorem 6.
As regards the assumptions of Theorem5, the constant factor k
(
λ
1
) in (26) and (27) is the best possible one.Proof.
For anyε
∈
(
0
,
q
λ
2
), we set λ
~
1
=
λ
1
+
ε
/
q, λ
~
2
=
λ
2
-
ε
/
q
(
∈
(
0,1
)
), and (34)
a
~
m
≔
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
1
-
ε
/
p
-
1
=
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
~
1
-
ε
-
1
m
∈
N
,
b
~
n
≔
n
-
η
+
n
-
η
cos
⁡
β
λ
2
-
ε
/
q
-
1
=
n
-
η
+
n
-
η
cos
⁡
β
λ
~
2
-
1
n
∈
N
.Then by (22) and (10), we find(35)
I
~
1
≔
∑
m
=
1
∞
m
-
ξ
+
m
-
ξ
cos
⁡
α
p
1
-
λ
1
-
1
a
~
m
p
1
/
p
∑
n
=
1
∞
n
-
η
+
n
-
η
cos
⁡
β
q
1
-
λ
2
-
1
b
~
n
q
1
/
q
=
∑
m
=
1
∞
1
m
-
ξ
+
m
-
ξ
cos
⁡
α
1
+
ε
1
/
p
∑
n
=
1
∞
1
n
-
η
+
n
-
η
cos
⁡
β
1
+
ε
1
/
q
=
1
ε
1
1
+
cos
⁡
α
1
+
ε
+
1
1
-
cos
⁡
α
1
+
ε
1
/
p
1
1
+
cos
⁡
β
1
+
ε
+
1
1
-
cos
⁡
β
1
+
ε
1
/
q
1
+
o
1
1
1
/
p
1
+
o
2
1
1
/
q
,
I
~
≔
∑
n
=
1
∞
∑
m
=
1
∞
k
m
,
n
a
~
m
b
~
n
=
∑
m
=
1
∞
∑
m
=
1
∞
k
m
,
n
m
-
ξ
+
m
-
ξ
cos
⁡
α
λ
~
1
-
ε
-
1
n
-
η
+
n
-
η
cos
⁡
β
1
-
λ
~
2
=
∑
m
=
1
∞
ω
λ
~
2
,
m
m
-
ξ
+
m
-
ξ
cos
⁡
α
ε
+
1
≥
k
β
λ
~
1
∑
m
=
1
∞
1
-
θ
λ
~
2
,
m
m
-
ξ
+
m
-
ξ
cos
⁡
α
ε
+
1
=
k
β
λ
~
1
∑
m
=
1
∞
1
m
-
ξ
+
m
-
ξ
cos
⁡
α
ε
+
1
-
∑
m
=
1
∞
1
O
m
-
ξ
+
m
-
ξ
cos
⁡
α
ε
/
p
+
λ
2
+
1
=
k
β
λ
~
1
ε
1
1
+
cos
⁡
α
1
+
ε
+
1
1
-
cos
⁡
α
1
+
ε
1
+
o
1
1
-
ε
O
1
.
If there exists a constantk
≤
k
(
λ
1
), such that (26) is valid when replacing k
(
λ
1
) by k, then, in particular, we have ε
I
~
<
k
I
~
1; namely,(36)
k
β
λ
~
1
1
1
+
cos
⁡
α
1
+
ε
+
1
1
-
cos
⁡
α
1
+
ε
1
+
o
1
1
-
ε
O
1
<
k
1
1
+
cos
⁡
α
1
+
ε
+
1
1
-
cos
⁡
α
1
+
ε
1
/
p
1
+
o
1
1
1
/
p
1
1
+
cos
⁡
β
1
+
ε
+
1
1
-
cos
⁡
β
1
+
ε
1
/
q
1
+
o
2
1
1
/
q
.It follows that(37)
4
B
λ
1
,
λ
2
csc
2
⁡
β
csc
2
⁡
α
≤
2
k
csc
2
/
p
⁡
α
csc
2
/
q
⁡
β
ε
⟶
0
+
;namely, k
(
λ
1
)
=
2
B
λ
1
,
λ
2
csc
2
/
p
⁡
β
csc
2
/
q
⁡
α
≤
k
. Hence, k
=
k
(
λ
1
) is the best possible constant factor of (26). The constant factor k
(
λ
1
) in (27) is still the best possible one. Otherwise, we would reach a contradiction by (30) that the constant factor in (26) is not the best possible one.We set functionsΦ
(
m
) and Ψ
(
n
) as follows:(38)
Φ
m
≔
m
-
ξ
+
m
-
ξ
cos
⁡
α
p
1
-
λ
1
-
1
m
∈
N
,
Ψ
n
≔
n
-
η
+
n
-
η
cos
⁡
β
q
1
-
λ
2
-
1
n
∈
N
,wherefrom Ψ
1
-
p
(
n
)
=
n
-
η
+
n
-
η
cos
⁡
β
p
λ
2
-
1
(
n
∈
N
)
. We also set the following weight normed spaces:(39)
l
p
,
Φ
≔
a
=
a
m
m
=
1
∞
;
a
p
,
Φ
=
∑
m
=
1
∞
Φ
m
a
m
p
1
/
p
<
∞
,
l
q
,
Ψ
≔
b
=
b
n
n
=
1
∞
;
b
q
,
Ψ
=
∑
n
=
1
∞
Ψ
n
b
n
q
1
/
q
<
∞
,
l
p
,
Ψ
1
-
p
≔
c
=
c
n
n
=
1
∞
;
c
p
,
Ψ
1
-
p
=
∑
n
=
1
∞
Ψ
1
-
p
n
c
n
p
1
/
p
<
∞
.Then, fora
=
a
m
m
=
1
∞
∈
l
p
,
Φ, c
=
c
n
n
=
1
∞, and c
n
=
∑
m
=
1
∞
k
(
m
,
n
)
a
m, in view of (27), we have c
p
,
Ψ
1
-
p
<
k
λ
1
a
p
,
Φ; namely, c
∈
l
p
,
Ψ
1
-
p
.Definition 7.
Define a Hilbert-type operatorT
:
l
p
,
Φ
→
l
p
,
Ψ
1
-
p as follows: for any a
=
a
m
m
=
1
∞
∈
l
p
,
Φ, there exists a unique representation c
=
T
a
∈
l
p
,
Ψ
1
-
p
. One also defines the formal inner product of T
a and b
=
b
n
n
=
1
∞
∈
l
q
,
Ψ
(
b
n
≥
0
) as follows:(40)
T
a
,
b
≔
∑
n
=
1
∞
∑
m
=
1
∞
k
m
,
n
a
m
b
n
.
Then fora
m
≥
0
(
m
∈
N
), we may rewrite (26) and (27) as follows:(41)
T
a
,
b
<
k
λ
1
a
p
,
Φ
b
q
,
Ψ
,
(42)
T
a
p
,
Ψ
1
-
p
<
k
λ
1
a
p
,
Φ
.
We define the norm of operatorT as follows:(43)
T
≔
sup
a
≠
θ
∈
l
p
,
Φ
⁡
T
a
p
,
Ψ
1
-
p
a
p
,
Φ
.Since, by Theorem 6, the constant factor k
(
λ
1
) in (42) is the best possible one, we have(44)
T
=
k
λ
1
=
2
B
λ
1
,
λ
2
csc
2
/
p
⁡
β
csc
2
/
q
⁡
α
.Remark 8.
(i) Forξ
=
η
=
0, (26) reduces to(45)
∑
n
=
1
∞
∑
m
=
1
∞
1
m
+
m
cos
⁡
α
+
n
+
n
cos
⁡
β
λ
a
m
b
n
<
k
λ
1
∑
m
=
1
∞
m
+
m
cos
⁡
α
p
1
-
λ
1
-
1
a
m
p
1
/
p
∑
n
=
1
∞
n
+
n
cos
⁡
β
q
1
-
λ
2
-
1
b
n
q
1
/
q
.Hence, (26) is a more accurate inequality of (45). In particular, for α
=
β
=
π
/
2 in (45), we have the following new inequality: (46)
∑
n
=
1
∞
∑
m
=
1
∞
a
m
b
n
m
+
n
λ
<
2
B
λ
1
,
λ
2
∑
m
=
1
∞
m
p
1
-
λ
1
-
1
a
m
p
1
/
p
∑
n
=
1
∞
n
q
1
-
λ
2
-
1
b
n
q
1
/
q
.
(ii) Forα
=
β
=
π
/
2 in (26), we have (5). For a
-
m
=
a
m and b
-
n
=
b
n
(
m
,
n
∈
N
), (5) reduces to(47)
∑
n
=
1
∞
∑
m
=
1
∞
1
m
+
n
-
ξ
-
η
λ
+
1
m
+
n
+
ξ
-
η
λ
+
1
m
+
n
+
ξ
+
η
λ
+
1
m
+
n
-
ξ
+
η
λ
a
m
b
n
<
2
B
λ
1
,
λ
2
∑
m
=
1
∞
m
-
ξ
p
1
-
λ
1
-
1
+
m
+
ξ
p
1
-
λ
1
-
1
a
m
p
1
/
p
∑
n
=
1
∞
n
-
η
q
1
-
λ
2
-
1
+
n
+
η
q
1
-
λ
2
-
1
b
n
q
1
/
q
.In particular, for ξ
=
η
∈
0
,
1
/
2, we have(48)
∑
n
=
1
∞
∑
m
=
1
∞
1
m
+
n
-
2
η
λ
+
2
m
+
n
λ
+
1
m
+
n
+
2
η
λ
a
m
b
n
<
2
B
λ
1
,
λ
2
∑
m
=
1
∞
m
-
η
p
1
-
λ
1
-
1
+
m
+
η
p
1
-
λ
1
-
1
a
m
p
1
/
p
∑
n
=
1
∞
n
-
η
q
1
-
λ
2
-
1
+
n
+
η
q
1
-
λ
2
-
1
b
n
q
1
/
q
.
Ifη
=
0, then (48) reduces to(49)
∑
n
=
1
∞
∑
m
=
1
∞
a
m
b
n
m
+
n
λ
<
B
λ
1
,
λ
2
∑
m
=
1
∞
m
p
1
-
λ
1
-
1
a
m
p
1
/
p
∑
n
=
1
∞
n
q
1
-
λ
2
-
1
b
n
q
1
/
q
,which is an extension of (1).



---

*Source: 9197476-2016-09-08.xml*