ISSN: 2008-6822 (electronic)
http://www.ijnaa.semnan.ac.ir
A Multidimensional Discrete Hilbert-Type Inequality
Bicheng Yang
Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P. R. China.
Dedicated to the Memory of Charalambos J. Papaioannou
(Communicated by Th. M. Rassias)
Abstract
In this paper, by using the way of weight coefficients and technique of real analysis, a multidimen-sional discrete Hilbert-type inequality with a best possible constant factor is given. The equivalent form, the operator expression with the norm are considered.
Foundation item: This work is supported by the National Natural Science Foundation of China (No.61370186), and 2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2012KJCX0079).
Keywords: Hilbert’s Inequality, Weight Coefficient, Equivalent Form, Operator, Norm.
2010 MSC: 26D15, 47A07.
1. Introduction
Assuming thatp > 1,1p+1q = 1, f(x), g(y)≥0, f ∈Lp(R
+), g∈Lq(R+),||f||p={
R∞
0 f
p(x)dx}1p >0,
||g||q >0, we have the following Hardy-Hilbert’s integral inequality (cf. [3]):
Z ∞
0
Z ∞
0
f(x)g(y)
x+y dxdy < π
sin(π/p)||f||p||g||q, (1.1)
with the best possible constant factor sin(π/p)π . If am, bn ≥ 0, a = {am}∞m=1 ∈ lp, b = {bn}∞n=1 ∈ lq,
||a||p ={
P∞
m=1a p m}
1
p >0,||b||
q >0,then we have the following discrete Hilbert’s inequality with the same best constant π
sin(π/p) :
∞
X
m=1
∞
X
n=1
ambn
m+n < π
sin(π/p)||a||p||b||q. (1.2)
Email address: [email protected]; [email protected] (Bicheng Yang)
Inequalities (1.1) and (1.2) are important in analysis and its applications (cf. [3], [11], [17], [14], [19], [20]).
In 1998, by introducing an independent parameter λ∈(0,1], Yang [18] gave an extension of (1.1) at p = q = 2. For improving the results of [18], Yang gave some extensions of (1.1) and (1.2) as follows (cf. [17]):
If λ1, λ2, λ∈R, λ1+λ2 =λ, kλ(x, y) is a non-negative homogeneous function of degree−λ, with
k(λ1) =
Z ∞
0
kλ(t,1)tλ1−1dt ∈R+,
φ(x) =xp(1−λ1)−1, ψ(x) = xq(1−λ2)−1, f(x), g(y)≥0,
f ∈Lp,φ(R+) =
f;||f||p,φ :={
Z ∞
0
φ(x)|f(x)|pdx}p1 <∞
,
g ∈Lq,ψ(R+),||f||p,φ,||g||q,ψ >0, then
Z ∞
0
Z ∞
0
kλ(x, y)f(x)g(y)dxdy < k(λ1)||f||p,φ||g||q,ψ, (1.3)
where the constant factor k(λ1) is the best possible. Moreover, if kλ(x, y) is finite and
kλ(x, y)xλ1−1(kλ(x, y)yλ2−1) is decreasing with respect to x >0(y >0),then for am,bn ≥0, a∈lp,φ =
n
a;||a||p,φ :={P
∞
n=1φ(n)|an|
p}1p <∞
o
,
b={bn}∞n=1 ∈lq,ψ,||a||p,φ,||b||q,ψ >0,we have (cf. [14])
∞
X
m=1
∞
X
n=1
kλ(m, n)ambn < k(λ1)||a||p,φ||b||q,ψ, (1.4)
with the best possible constant factor k(λ1).
Clearly, for λ = 1, k1(x, y) = x+y1 , λ1 = 1q, λ2 = 1p, (1.3) reduces to (1.1), while (1.4) reduces to (1.2). Some other results including multidimensional Hilbert-type inequalities are provided by [23]-[10].
On half-discrete Hilbert-type inequalities with the non-homogeneous kernels, Hardy et al. pro-vided a few results in Theorem 351 of [3]. But they did not prove that the the constant factors are the best possible. However, Yang [21] gave a result with the kernel 1
(1+nx)λby introducing a variable and
proved that the constant factor is the best possible. In 2011 Yang [22] gave the following half-discrete Hardy-Hilbert’s inequality with the best possible constant factorB(λ1, λ2):
Z ∞
0
f(x)
∞
X
n=1
an
(x+n)λdx < B(λ1, λ2)||f||p,φ||a||q,ψ, (1.5)
whereλ1, λ2 >0, 0≤λ2 ≤1,λ1+λ2 =λ,B(u, v) =
R∞
0 1 (1+t)u+vt
u−1dt(u, v >0)is the beta function. Zhong et al ([29]–[37]) investigated several half-discrete Hilbert-type inequalities with particular kernels.
Applying the way of weight functions and the techniques of discrete and integral Hilbert-type inequalities with some additional conditions on the kernel, a half-discrete Hilbert-type inequality with a general homogeneous kernel of degree −λ ∈ R and a best constant factor k(λ1) is obtained as follows:
Z ∞
0
f(x)
∞
X
n=1
which is an extension of (1.5) (see Yang and Chen [24]). At the same time, a half-discrete Hilbert-type inequality with a general non-homogeneous kernel and a best constant factor is given by Yang [15].
In this paper, by using the way of weight coefficients and technique of real analysis, a multi-dimensional discrete Hilbert-type inequality with parameters and a best possible constant factor is given, which is a multidimensional extension of (1.4) for kλ(m, n) = ln(m/n)mλ−nλ. The equivalent form,
the operator expression with the norm are also considered.
2. Some lemmas
If i0, j0 ∈N(Nis the set of positive integers), α, β >0,we put
||x||α : = i0
X
k=1
|xk|α
!α1
(x= (x1,· · ·, xi0)∈R
i0), (2.1)
||y||β : = j0
X
k=1
|yk|β
!1β
(y= (y1,· · ·, yj0)∈R
j0). (2.2)
Lemma 2.1. If s ∈ N,γ, M >0,Ψ(u) is a non-negative measurable function in (0,1], and DM :=
x∈Rs +;
Ps
i=1x γ
i ≤Mγ ,then we have (cf. [16])
Z
· · ·
Z
DM
Ψ s
X
i=1
xi
M
γ !
dx1· · ·dxs (2.3)
= M
sΓs(1 γ)
γsΓ(s γ)
Z 1
0
Ψ(u)uγs−1du. (2.4)
Lemma 2.2. For s∈N,γ >0, ε >0, we have
X
m
||m||−s−ε
γ =
Γs(γ1)
εsε/γγs−1Γ(s γ)
+O(1)(ε→0+). (2.5)
Proof . ForM > s1/γ,we set Ψ(u) =
0,0< u < Msγ,
(M u1/γ)−s−ε, s
Mγ ≤u≤1.
Then by the decreasing property
and (2.4), it follows P
m||m||
−s−ε
γ ≥
R
{x∈Rs
+;xi≥1}||x||
−s−ε γ dx
= lim M→∞
Z
· · ·
Z
DM
Ψ s
X
i=1
xi
M
γ !
dx1· · ·dxs (2.6)
= lim M→∞
MsΓs(1γ)
γsΓ(s γ)
Z 1
s/Mγ
(M u1/γ)−s−εuγs−1du=
Γs(1γ)
εsε/γγs−1Γ(s γ)
, (2.7)
X
m
||m||−γs−ε = a+ X
{m∈Ns;m i≥2}
||m||−γs−ε
≤ a+
Z
{u∈Rs
+;ui≥1}
||u||−s−ε
γ du=a+
Γs(1 γ)
εsε/γγs−1Γ(s γ)
.
Definition 2.3. For α, β >0, λ > 0,0< λ1 ≤i0,0 < λ2 ≤ j0, λ1+λ2 =λ, m= (m1,· · ·, mi0)∈ Ni0, n= (n
1,· · ·, nj0)∈N
j0, define two weight coefficients w
λ(λ2, n) and Wλ(λ1, m) as follows:
wλ(λ2, n) : =
X
m
ln(||m||α/||n||β)
||m||λ
α− ||n||λβ
||n||λ2 β
||m||i0−λ1 α
, (2.8)
Wλ(λ1, m) : =
X
n
ln(||m||α/||n||β)
||m||λ
α− ||n||λβ
||m||λ1 α
||n||j0−λ2 β
, (2.9)
where, P
m =
P∞
mi0=1· · ·
P∞
m1=1 and
P
n=
P∞
nj0=1· · ·
P∞
n1=1 . Lemma 2.4. As the assumptions of Definition 1, then (i) we have
wλ(λ2, n) < K2(n ∈Nj0), (2.10)
Wλ(λ1, m) < K1(m∈Ni0), (2.11)
where,
K1 =
Γj0(1 β)
βj0−1Γ(j0 β)
"
π λsin(πλ1
λ )
#2
, (2.12)
K2 =
Γi0(1 α)
αi0−1Γ(i0 α)
"
π λsin(πλ1
λ )
#2
; (2.13)
(ii) for p > 1, 0< ε < pλ1,setting eλ1 =λ1− εp,eλ2 =λ2+ εp, we have
0<Ke2(1−eθλ(n))< wλ(eλ2, n), (2.14)
where,
e
θλ(n) =
"
1
πsin( πeλ1
λ )
#2
Z iλ/α0 /||n||λβ
0
(lnv)v(eλ1/λ)−1
v−1 dv (2.15)
= O
1
||n||λe1−λ21 β
, (2.16)
e
K2 =
Γi0(1 α)
αi0−1Γ(i0 α)
"
π
λsin(πeλ1 λ )
#2
. (2.17)
Proof . Proof. By the decreasing property and (2.4), it follows
wλ(λ2, n)<
Z
Ri0 +
ln(||x||α/||n||β)
||x||λ
α− ||n||λβ
||n||λ2 β
||x||i0−λ1 α
dx
= lim M→∞
Z
DM
ln(M[Pi0 i=1(
xi
M)
α]α1/||n|| β)
Mλ[Pi0 i=1(
xi
M)α]
λ
α − ||n||λ
β
||n||λ2
β dx1· · ·dxi0
Mi0−λ1[Pj0 i=1(
xi
M)α]
i0−λ1
α
= lim M→∞
Mi0Γi0(1 α)
αi0Γ(i0 α)
Z 1 0
ln(M uα1/||n||β)
Mλuλα − ||n||λ
β
||n||λ2 β u
i0 α−1
Mi0−λ1u
i0−λ1
α
= lim M→∞
Mλ1Γi0(1 α)
αi0Γ(i0 α)
Z 1
0
ln(M u1α/||n||β)||n||λ2
β
Mλuλα − ||n||λ
β
uλα1−1du
u=||n−σ||α βM
−αvα/λ= Γ i0(1
α)
λ2αi0−1Γ(i0 α)
Z ∞
0
(lnv)v(λ1/λ)−1
v −1 dv
= Γ
i0(1 α)
αi0−1Γ(i0 α)
"
π λsin(πλ1
λ )
#2
=K2.
Hence, we have (2.10). By the same way, we have (2.11). Since limv→0+ v
λ1/(2λ)lnv
v−1 = 0, there exists a constantM > 0,such that
vλ1/(2λ)lnv
v−1 ≤M(v ∈(0, i
λ(α1−1
β)
0 ]).
By the decreasing property and the same way as obtaining (2.7), we have
wλ(eλ2, n)> Z
{x∈Ri0 +;xi≥1}
ln(||x||α/||n||β)
||x||λ
α− ||n||λβ
||n||eλ2 β
||x||i0−eλ1 α
dx
= Γ
i0(1 α)
λ2αi0−1Γ(i0 α)
Z ∞
iλ/α0 /||n||λ β
(lnv)v(eλ1/λ)−1
v−1 dv=Ke2(1−θeλ(n))>0,
0 < θeλ(n) = "
1
πsin( πeλ1
λ )
#2
Z iλ/α0 /||n||λβ
0
(lnv)v(eλ1/λ)−1dv
v−1
≤
"
1
πsin( πeλ1
λ )
#2
M
Z iλ/α0 /||n||λ β
0
veλλ1− λ1
2λ−1dv
= λMsin 2(πeλ1
λ ) (eλ1− λ21)π2
i(eλ1−λ21)/α 0
||n||λe1−λ21 β
.
The lemma is proved.
3. Main Results and Operator Expressions
Setting Φ(m) := ||m||p(i0−λ1)−i0
α (m ∈Ni0) and Ψ(n) :=||n||q(j0
−λ2)−j0
β (n∈N
j0), we have
Theorem 3.1. If α, β > 0, λ >0,0< λ1 ≤ i0,0< λ2 ≤j0, λ1+λ2 =λ, then for p > 1,1p +1q = 1
am, bn ≥0,0<||a||p,Φ,||b||q,Ψ<∞, we have the following inequality
I :=X
n
X
m
ln(||m||α/||n||β)
||m||λ
α− ||n||λβ
ambn< K 1
p
1K 1
q
2||a||p,Φ||b||q,Ψ, (3.1)
where the constant factor
K
1
p
1K 1
q
2 =
"
Γj0(1 β)
βj0−1Γ(j0 β)
#1p
Γi0(1 α)
βi0−1Γ(i0 α)
1q " π
λsin(πλ1 λ )
#2
(3.2)
Proof . By H¨older’s inequality (cf. [8]), we have
I =X
n
X
m
ln(||m||α/||n||β)
||m||λ
α− ||n||λβ
"
||m||(i0−λ1)/q α
||n||(j0−λ2)/p β
am
# "
||n||(j0−λ2)/p β
||m||(i0−λ1)/q α bn # ≤ ( X m
Wλ(λ1, m)||m||αp(i0−λ1)−i0a p m
)1p
×
( X
n
wλ(λ2, n)||n||
q(j0−λ2)−j0
β b
q n
)1q
.
Then by (2.10) and (2.11), we have (3.1).
For 0< ε < pλ1, eλ1 =λ1− pε,eλ2 =λ2+pε,we set
eam =||m||
−i0+λ1−εp
α ,ebn =||n||
−j0+λ2−εq
β (m∈N
i0, n ∈Nj0).
Then by (2.5) and (2.14), we obtain
||ea||p,Φ||eb||q,Ψ= (
X
m
||m||p(i0−λ1)−i0
α ea
p m
)1p ( X
n
||n||q(j0−λ2)−j0 β ebqn
)1q
=
( X
m
||m||−i0−ε α
)1p( X
n
||n||−j0−ε β
)1q
(3.3)
= 1
ε
"
Γi0(1 α)
iε/α0 αi0−1Γ(i0 α)
+εO(1)
#1p
(3.4)
×
"
Γj0(1 β)
j0ε/ββj0−1Γ(j0 β)
+εOe(1) #1q
, (3.5)
e
I : =X
n
" X
m
ln(||m||α/||n||β)
||m||λ
α− ||n||λβ
e
am
#
ebn = X
n
wλ(eλ2, n)||n||
−j0−ε
β (3.6)
> Ke2 X
n
1−O(
1
||n||eλ1−
λ1 2 β ) ||n||
−j0−ε
β (3.7)
= Ke2 "
Γj0(1 β)
εj0ε/ββj0−1Γ(j0 β)
+Oe(1)−O(1) #
. (3.8)
If there exists a constant K ≤ K
1
p
1K 1
q
2, such that (3.1) is valid as we replace K 1
p
1K 1
q
2 by K, then we have Ke2
Γj0(1
β)
jε/β0 βj0−1Γ(
j0 β)
+εOe(1)−εO(1)
< εIe
< εK||ea||p,ϕ||eb||q,ψ =K "
Γi0(1 α)
iε/α0 αi0−1Γ(i0 α)
+εO(1)
#1p
×
"
Γj0(1 β)
j0ε/ββj0−1Γ(j0 β)
+εOe(1) #1q
Forε→0+,we find Γj0(1 β)
βj0−1Γ(j0
β)
Γi0 ( 1α)
αi0−1Γ(i0
α)
[ π
λsin(πλ1
λ )
]2≤K[ Γi0 ( 1α) αi0−1Γ(i0
α)
] 1
p[ Γ j0 ( 1
β) βj0−1Γ(j0
β)
] 1
q,
and thenK
1
p
1K 1
q
2 ≤K.
Hence, K =K
1
p
1K 1
q
2 is the best possible constant factor of (3.1).
Theorem 3.2. As the assumptions of Theorem 1, for 0 < ||a||p,Φ < ∞, we have the following
inequality with the best constant factor K
1
p
1K 1
q
2 :
J :=
( X
n
||n||pλ2−j0 β
X
m
ln(||m||α/||n||β)am
||m||λ
α− ||n||λβ
!p)1p
< K
1
p
1K 1
q
2||a||p,Φ, (3.9)
which is equivalent to (3.1).
Proof . We set bn as follows: bn := ||n||pλ2
−j0 β
P
m
ln(||m||α/||n||β)am
||m||λ α−||n||λβ
p−1
, n ∈ Nj0.Then it follows
Jp = ||b||q
q,Ψ. If J = 0, then (3.9) is trivially valid for 0 < ||a||p,Φ < ∞; if J = ∞, then it is impossible since the right hand side of (3.9) is finite. Suppose that 0 < J <∞. Then by (3.1), we find ——b——qq,Ψ = Jp = I < K
1
p
1 K 1
q
2 ||a||p,Φ||b||q,Ψ,namely, ||b||q
−1
q,Ψ = J < K 1
p
1 K 1
q
2||a||p,Φ, and then (3.9) follows.
On the other hand, assuming that (3.9) is valid, by H¨older’s inequality, we have
I =X
n
(Ψ(n))−q1
" X
m
ln(||m||α/||n||β)am
||m||λ
α− ||n||λβ
#
[(Ψ(n))1qbn]≤J||b||
q,Ψ. (3.10)
Then by (3.9), we have (3.1). Hence (3.9) and (3.1) are equivalent. By the equivalency, the constant factor K
1
p
1 K 1
q
2 in (3.9) is the best possible. Otherwise, we can come to a contradiction by (3.10) that the constant factor K
1
p
1 K 1
q
2 in (3.1) is not the best possible.
Forp > 1, we define two real weight normal discrete spaces lp,ϕ and lq,ψ as follows:
lp,ϕ : =
(
a={am};||a||p,Φ ={
X
m
Φ(m)apm}1p <∞
)
,
lq,ψ : =
(
b ={bn};||b||q,Ψ ={
X
n
Ψ(n)bqn}1q <∞
)
.
As the assumptions of Theorem 1, in view of J < K
1
p
1K 1
q
2||a||p,Φ,we have the following definition: Definition 3.3. Define a multidimensional Hilbert-type operator T :lp,Φ → lp,Ψ1−p as follows: For
a∈lp,Φ, there exists an unique representation T a∈lp,Ψ1−p, satisfying
T a(n) :=X m
ln(||m||α/||n||β)am
||m||λ
α− ||n||λβ
(n∈Nj0). (3.11)
For b∈lq,Ψ, we define the following formal inner product of T a and b as follows:
(T a, b) :=X n
X
m
ln(||m||α/||n||β)am
||m||λ
α− ||n||λβ
Then by Theorem 1 and Theorem 2, for 0<||a||p,ϕ,||b||q,ψ <∞,we have the following equivalent inequalities:
(T a, b) < K
1
p
1 K 1
q
2||a||p,Φ||b||q,Ψ, (3.13)
||T a||p,Ψ1−p < K
1
p
1 K 1
q
2||a||p,Φ. (3.14)
It follows thatT is bounded with
||T||:= sup a(6=θ)∈lp,Φ
||T a||p,Ψ1−p
||a||p,Φ
≤K
1
p
1 K 1
q
2. (3.15)
Since the constant factor K
1
p
1K 1
q
2 in (3.14) is the best possible, we have Corollary 3.4. If T is defined by Definition 2, then it follows
||T||=K
1
p
1K 1
q
2 = [
Γj0(1 β)
βj0−1Γ(j0 β)
]1p[ Γ
i0(1 α)
αi0−1Γ(i0 α)
]1q[ π
λsin(πλ1 λ )
]2. (3.16)
Remark 3.5. For i0 =j0 = 1 in (3.1), we have inequality
∞
X
m=1
∞
X
n=1
ln(m/n)
mλ−nλambn <
"
π λsin(πλ1
λ )
#2
||a||p,φ||b||q,ψ. (3.17)
Hence, (3.1) is a multidimensional extension of (1.4) for kλ(m, n) = ln(m/n)mλ−nλ.
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[20] B. C. Yang,Two Types of Multiple Half-Discrete Hilbert-Type Inequalities, Lambert Academic Publishing, 2012. [21] B. C. Yang,A mixed Hilbert-type inequality with a best constant factor, International Journal of Pure and Applied
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[29] W. Y. Zhong, A mixed Hilbert-type inequality and its equivalent forms Journal of Guangdong University of Education, 31 (5) (2011) 18-22.
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[31] J. H. Zhong,Two classes of half-discrete reverse Hilbert-type inequalities with a non-homogeneous kernel, Journal of Guangdong University of Education, 32 (5) (2012) 11-20.
[32] J. H. Zhong,A half-discrete Hilbert-type inequality, Journal of Zhejiang University (Science Edition),40 (1) (2013) 19-22.
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[34] J. H. Zhong and B. C. Yang, On an extension of a more accurate Hilbert-type inequality, Journal of Zhejiang University (Science Edition), 35 (2) (2008) 121-124.
[35] W. Y. Zhong and B. C. Yang,A best extension of Hilbert inequality involving several parameters, Journal of Jinan University (Natural Science), 28 (1) (2007) 20-23.
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