R E S E A R C H
Open Access
A Hilbert-type integral inequality in the
whole plane with a non-homogeneous kernel
and a few parameters
Zhaohui Gu
1*and Bicheng Yang
2*Correspondence:
[email protected] 1School of Economics and Trade,
Guangdong University of Foreign Studies, Guangzhou, Guangdong 510006, P.R. China
Full list of author information is available at the end of the article
Abstract
By using the way of real analysis and estimating the weight functions, we build a new Hilbert-type integral inequality in the whole plane with a non-homogeneous kernel and a few parameters. The constant factor related to the beta function is proved to be the best possible. We also consider the equivalent forms, the reverses, and some particular cases.
MSC: 26D15
Keywords: Hilbert-type integral inequality; weight function; equivalent form; beta function; reverse
1 Introduction
Iff(x),g(y)≥, satisfying <∞f(x)dx<∞and <∞
g(y)dy<∞, then we have (cf.[])
∞
∞
f(x)g(y)
x+y dx dy<π
∞
f(x)dx
∞
g(y)dy
, ()
where the constant factorπis the best possible. Inequality () is known as Hilbert’s integral inequality, which is important in analysis and its applications (cf.[, ]).
In recent years, by using the way of weight functions, a number of extensions of () were given by Yang (cf.[]). Noticing that inequality () is a homogeneous kernel of degree –, in , A survey of the study of Hilbert-type inequalities with the homogeneous kernels of degree negative numbers and some parameters is given by []. Recently, some inequal-ities with the homogeneous kernels of degree and non-homogeneous kernels have been studied (cf.[–]). All of the above integral inequalities are built in the quarter plane.
In , Yang [] first gave a Hilbert-type integral inequality in the whole plane as fol-lows:
∞
–∞
∞
–∞
f(x)g(y) ( +ex+y)λdx dy
<B
λ ,
λ
∞
–∞
e–λxf(x)dx
∞
–∞
e–λyg(y)dy
, ()
where the constant factorB(λ,λ) (λ> ) is the best possible, and
B(u,v) :=
∞
tu+
( +t)u+vdt (u,v> ) ()
is the beta function (cf.[]). Heet al.[–] also provided some Hilbert-type integral inequalities in the whole plane.
In this paper, by using the way of real analysis and estimating the weight functions, we build a new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and a few parameters. The constant factor related to the beta function is proved to be the best possible. We also consider the equivalent forms, the reverses, and some particular cases.
2 Some lemmas
Lemma Suppose that <α≤α<π,μ,σ > ,μ+σ =λ,γ ∈ {k+, k– (k∈N)},
δ∈ {–, }.We define weight functionsω(σ,y) (y∈R),and (σ,x) (x∈R)as follows:
ω(σ,y) :=
∞
–∞
min i∈{,}
|y|σ|x|δσ– [|xδy|γ + (xδy)γcosα
i+ ]λ/γ
dx, ()
(σ,x) :=
∞
–∞
min i∈{,}
|x|δσ|y|σ– [|xδy|γ+ (xδy)γcosα
i+ ]λ/γ
dy. ()
Then for y,x∈R\{},we have
ω(σ,y) = (σ,x) =K(σ)
:= γσ/γ
secα
σ γ
+
cscα
σ γ
B
μ γ,
σ γ
∈R+. ()
Proof (i) Forδ= ,y∈R\{}, settingu=xy, we find
ω(σ,y) =
∞
–∞
min i∈{,}
(|u|γ+uγcosα
i+ )λ/γ| u|σ–du
=
∞
min i∈{,}
[uγ( +cosα
i) + ]λ/γ uσ–du
+
–∞
min i∈{,}
[uγ(– +cosα
i) + ]λ/γ
(–u)σ–du
=
∞
min i∈{,}
[uγ( +cosα
i) + ]λ/γ uσ–du
+
∞
min i∈{,}
[vγ( –cosα
i) + ]λ/γ vσ–dv
=
∞
uσ–du [uγ( +cosα
) + ]λ/γ +
∞
vσ–dv [vγ( –cosα
) + ]λ/γ
. ()
Settingt=uγ( +cosα
) (t=uγ( –cosα)) in the above first (second) integral, by (), it follows that
ω(σ,y) = γ
secα
σ γ
+
cscα
σ
γ ∞
tσγ–dt
(ii) Forδ= –, setting yx, we still can obtainω(σ,y) =K(σ). Settingu=xδy, we also find
(σ,x) =
∞
–∞
min i∈{,}
(|u|γ+uγcosα
i+ )λ/γ|
u|σ–du=K(σ).
Hence we have ().
Note If we replacemini∈{,}bymaxi∈{,}in () and (), then we may exchangeαandα in ().
Lemma Suppose that p> ,p+q= , <α≤α<π,μ,σ> ,μ+σ=λ,γ∈ {k+, k– (k∈N)},δ∈ {–, }.If K(σ)is indicated by(),f(x)is a non-negative measurable function in(–∞,∞),then we have
J:=
∞
–∞|y|
pσ–
∞
–∞
min i∈{,}
[|xδy|γ+ (xδy)γcosα
i+ ]λ/γ f(x)dx
p dy
≤Kp(σ)
∞
–∞|x|
p(–δσ)–fp(x)dx. ()
Proof We set
kλ(δ)(x,y) := min i∈{,}
[|xδy|γ+ (xδy)γcosα
i+ ]λ/γ
(x,y∈R). ()
By Hölder’s inequality (cf.[]), we have
∞
–∞k
(δ)
λ (x,y)f(x)dx
p
=
∞
–∞k
(δ)
λ (x,y)
|
x|(–δσ)/q
|y|(–σ)/pf(x)
|
y|(–σ)/p
|x|(–δσ)/q
dx p
≤
∞
–∞
k(λδ)(x,y)|x|
(–δσ)(p–)
|y|–σ f p(x)dx
×
∞
–∞
kλ(δ)(x,y)
|y|(–σ)(q–) |x|–δσ dx
p–
=ω(σ,y)p–|y|–pσ+
∞
–∞
kλ(δ)(x,y)
|x|(–δσ)(p–) |y|(–σ) f
p(x)dx. ()
Then by () and the Fubini theorem (cf.[]), it follows that
J≤Kp–(σ)
∞
–∞
∞
–∞k
(δ)
λ (x,y)
|x|(–δσ)(p–) |y|(–σ) f
p(x)dx
dy
=Kp–(σ)
∞
–∞ (σ,x)|x|
p(–δσ)–fp(x)dx.
Hence, still in view of (), inequality () follows.
3 Main results and applications
Theorem Suppose that p> , p + q = , <α ≤α<π, μ,σ > , μ+σ =λ, γ ∈ {
<–∞∞|x|p(–δσ)–fp(x)dx<∞and <∞
–∞|y|q(–
σ)–gq(y)dy<∞,then we have the
fol-lowing equivalent inequalities:
I:=
∞
–∞
∞
–∞
min i∈{,}
[|xδy|γ+ (xδy)γcosα
i+ ]λ/γ
f(x)g(y)dx dy
<K(σ)
∞
–∞|x|
p(–δσ)–fp(x)dx
p ∞
–∞|y|
q(–σ)–gq(y)dy
q
, ()
J:=
∞
–∞ |y|pσ–
∞
–∞
min i∈{,}
[|xδy|γ+ (xδy)γcosα
i+ ]λ/γ f(x)dx
p dy
<Kp(σ)
∞
–∞
|x|p(–δσ)–fp(x)dx, ()
where the constant factors K(σ)and Kp(σ)are the best possible. In particular,forα=α=α∈(,π),γ = in()and(),we find
K(σ) =k(σ) := σ
secα
σ
+
cscα
σ
B(μ,σ), ()
and the following equivalent inequalities:
∞
–∞
∞
–∞
(|xδy|+xδycosα+ )λf(x)g(y)dx dy
<k(σ)
∞
–∞
|x|p(–δσ)–fp(x)dx
p ∞
–∞
|y|q(–σ)–gq(y)dy
q
, ()
∞
–∞|y|
pσ–
∞
–∞
(|xδy|+xδycosα+ )λf(x)dx
p dy
<kp(σ)
∞
–∞|x|
p(–σ)–fp(x)dx. ()
Proof If () takes the form of equality fory∈(–∞, )∪(,∞), then there exist constants
AandB, such that they are not all zero, and
A|x|
(–δσ)(p–)
|y|(–σ) f
p(x) =B|y|(–σ)(q–)
|x|(–δσ) a.e. in (–∞,∞).
We supposeA= (otherwiseB=A= ). Then it follows that
|x|p(–δσ)–fp(x) =|y|q(–σ) B
A|x| a.e. in (–∞,∞),
which contradicts the fact that <–∞∞|x|p(–δσ)–fp(x)dx<∞. Hence () takes the form
of a strict inequality. So does (), and we have (). By Hölder’s inequality (cf.[]), we find
I =
∞
–∞
|y|σ–p
∞
–∞k
(δ)
λ (x,y)f(x)dx
|y|p–σg(y)dy
≤Jp
∞
–∞
|y|q(–σ)–gq(y)dy
q
Then by (), we have (). On the other hand, suppose that () is valid. Setting
g(y) :=|y|pσ–
∞
–∞k
(δ)
λ (x,y)f(x)dx
p–
, y∈R,
then it follows thatJ=–∞∞|y|q(–σ)–gq(y)dy. By (), we haveJ<∞. IfJ= , then () is
obviously of value; if <J<∞, then by (), we obtain
∞
–∞
|y|q(–σ)–gq(y)dy
=J=I<K(σ)
∞
–∞|x|
p(–δσ)–fp(x)dx
p ∞
–∞|y|
q(–σ)–gq(y)dy
q
, ()
Jp=
∞
–∞
|y|q(–σ)–gq(y)dy
p
<K(σ)
∞
–∞|x|
p(–δσ)–fp(x)dx
p
. ()
Hence we have (), which is equivalent to (). We setEδ:={x∈R;|x|δ≥}, andE+
δ :=Eδ∩R+={x∈R+;x
δ≥}. Forε> , we define
functionsf˜(x),g˜(y) as follows:
˜
f(x) :=
|x|δ(σ–pε)–, x∈Eδ, , x∈R\Eδ,
˜
g(y) :=
, y∈(–∞, –)∪(,∞), |y|σ+qε–, y∈[–, ].
Then we obtain
˜
L:=
∞
–∞
|x|p(–δσ)–f˜p(x)dx
p ∞
–∞
|y|q(–σ)–g˜q(y)dy
q
=
Eδ+
x–δε–dx
p
yε–dy
q
= ε.
We find
h(x) :=
–
min i∈{,}
|y|σ+qε– [|xδy|γ + (xδy)γcosα
i+ ]λ/γ
dy=h(–x).
In fact, settingY= –y, we obtain
h(–x) =
–
min i∈{,}
|y|σ+qε– [|–xδy|γ+ (–xδy)γcosα
i+ ]λ/γ dy
=
–
min i∈{,}
|Y|σ+qε– [|xδY|γ+ (xδY)γcosα
i+ ]λ/γ
It follows that ˜ I = ∞ –∞ ∞
–∞k
(δ)
λ (x,y)f˜(x)g˜(y)dx dy
=
Eδ
|x|δ(σ–pε)–h(x)dx=
E+
δ
xδ(σ–pε)–h(x)dx
u=xδy
=
E+δ x–δε–
xδ
–xδ min i∈{,}
|u|σ+qε– [|u|γ +uγcosα
i+ ]λ/γ
du dx.
Settingv=xδin the above integral, by the Fubini theorem (cf.[]), we find
˜
I=
∞
v–ε–
v
–v min i∈{,}
|u|σ+qε– [|u|γ +uγcosα
i+ ]λ/γ du dv
=
∞
v–ε–
v
min i∈{,}
[uγ( +cosα
i) + ]λ/γ
+min i∈{,}
[uγ( –cosα
i) + ]λ/γ
uσ+qε–du dv
=
∞
v–ε–
v
[uγ( +cosα
) + ]λ/γ
+
[uγ( –cosα
) + ]λ/γ
uσ+qε–du dv
=
∞
v–ε–
uσ+qε–
[uγ( +cosα
) + ]
λ γ
+ u
σ+qε–
[uγ( –cosα
) + ] λ γ du dv + ∞
v–ε–
v
uσ+qε–
[uγ( +cosα
) + ]
λ γ
+ u
σ+qε–
[uγ( –cosα
) + ] λ γ du dv = ε
uσ+qε– [uγ( +cosα
) + ]λ/γ
+ u
σ+qε–
[uγ( –cosα
) + ]λ/γ
du + ∞ ∞ u
v–ε–dv
×
uσ+qε– [uγ( +cosα
) + ]λ/γ
+ u
σ+qε–
[uγ( –cosα
) + ]λ/γ
du = ε
uσ+qε– [uγ( +cosα
) + ]λ/γ
+ u
σ+qε–
[uγ( –cosα
) + ]λ/γ
du + ∞
uσ–pε– [uγ( +cosα
) + ]λ/γ
+ u
σ–ε
p–
[uγ( –cosα
) + ]λ/γ
du .
If the constant factorK(σ) in () is not the best possible, then there exists a positive numberk, withK(σ) <k, such that () is valid when replacingK(σ) byk. Then we have εI˜<εkL˜, and
uσ+qε– [uγ( +cosα
) + ]λ/γ
+ u
σ+ε
q–
[uγ( –cosα
) + ]λ/γ
du
+
∞
uσ–pε– [uγ( +cosα
) + ]λ/γ
+ u
σ–pε–
[uγ( –cosα
) + ]λ/γ
du
By () and the Levi theorem (cf.[]), we have
K(σ) =
∞
uσ–du [uγ( +cosα
) + ]λ/γ +
∞
uσ–du [uγ( –cosα
) + ]λ/γ
=
lim ε→+
uσ+qε– [uγ( +cosα
) + ]λ/γ
+ u
σ+ε
q–
[uγ( –cosα
) + ]λ/γ
du
+
∞
lim ε→+
uσ–pε– [uγ( +cosα
) + ]λ/γ
+ u
σ–pε–
[uγ( –cosα
) + ]λ/γ
du
= lim ε→+
uσ+qε– [uγ( +cosα
) + ]λ/γ
+ u
σ+qε–
[uγ( –cosα
) + ]λ/γ
du
+
∞
uσ–pε– [uγ( +cosα
) + ]λ/γ
+ u
σ–ε
p–
[uγ( –cosα
) + ]λ/γ
du ≤k,
which contradicts the fact thatk<K(σ). Hence the constant factorK(σ) in () is the best possible.
If the constant factor in () is not the best possible, then by (), we may get a contra-diction: that the constant factor in () is not the best possible.
Theorem As the assumptions of Theorem,replacing p> by <p< ,we have the equivalent reverses of()and()with the same best constant factors.
Proof By the reverse Hölder’s inequality (cf.[]), we have the reverses of () and (). It is easy to obtain the reverse of (). In view of the reverses of () and (), we ob-tain the reverse of (). On the other hand, suppose that the reverse of () is valid. Set-ting the sameg(y) as Theorem , by the reverse of (), we haveJ> . IfJ=∞, then the reverse of () is obviously value; ifJ<∞, then by the reverse of (), we obtain the re-verses of () and (). Hence we have the reverse of (), which is equivalent to the reverse of ().
If the constant factorK(σ) in the reverse of () is not the best possible, then there exists a positive constantk, withk>K(σ), such that the reverse of () is still valid when replacing
K(σ) byk. By the reverse of (), we have
uσ+qε– [uγ( +cosα
) + ]λ/γ
+ u
σ+qε–
[uγ( –cosα
) + ]λ/γ
du
+
∞
uσ–pε– [uγ( +cosα
) + ]λ/γ
+ u
σ–pε–
[uγ( –cosα
) + ]λ/γ
du>k. ()
Forε→+, by the Levi theorem (cf.[]), we find that
∞
uσ–pε– [uγ( +cosα
) + ]λ/γ
+ u
σ–pε–
[uγ( –cosα
) + ]λ/γ
du
→
∞
uσ– [uγ( +cosα
) + ]λ/γ
+ u
σ–
[uγ( –cosα
) + ]λ/γ
There exists a constantδ> , such thatσ–δ> , and thenK(σ–δ) <∞. For <
ε<δ|q|(q< ), sinceuσ+qε–≤uσ–δ
–,u∈(, ], and
<
uσ–δ –
[uγ( +cosα
) + ]λ/γ
+ u
σ–δ –
[uγ( –cosα
) + ]λ/γ
du
≤K
σ–δ
<∞,
then by the Lebesgue control convergence theorem (cf.[]), forε→+, we have
uσ+qε– [uγ( +cosα
) + ]λ/γ
+ u
σ+qε–
[uγ( –cosα
) + ]λ/γ
du
→
uσ– [uγ( +cosα
) + ]λ/γ
+ u
σ–
[uγ( –cosα
) + ]λ/γ
du. ()
By (), (), and (), forε→+, we findK(σ)≥k, which contradicts the fact thatk>
K(σ). Hence, the constant factorK(σ) in the reverse of () is the best possible.
If the constant factor in reverse of () is not the best possible, then by the reverse of (), we may get a contradiction that the constant factor in the reverse of () is not the
best possible.
Remarks Forδ= – in () and (), replacing|x|λf(x) byf(x), we obtain the following
equivalent inequalities with the homogeneous kernel and the best possible constant fac-tors:
∞
–∞
∞
–∞
min i∈{,}
(|y|γ+sgn(x)yγcosα
i+|x|γ)λ/γ
f(x)g(y)dx dy
<K(σ)
∞
–∞
|x|p(–μ)–fp(x)dx
p ∞
–∞
|y|q(–σ)–gq(y)dy
q
, ()
∞
–∞|y|
pσ–
∞
–∞
min i∈{,}
(|y|γ +sgn(x)yγcosα
i+|x|γ)λ/γ f(x)dx
p dy
<Kp(σ)
∞
–∞|x|
p(–μ)–fp(x)dx. ()
In particular, forα=α=α∈(,π), γ = in () and (), we obtain the following equivalent inequalities:
∞
–∞
∞
–∞
(|y|+sgn(x)ycosα+|x|)λf(x)g(y)dx dy
<k(σ)
∞
–∞
|x|p(–μ)–fp(x)dx
p ∞
–∞
|y|q(–σ)–gq(y)dy
q
, ()
∞
–∞|y|
pσ–
∞
–∞
(|y|+sgn(x)ycosα+|x|)λf(x)dx
p dy
<kp(σ)
∞
–∞
|x|p(–μ)–fp(x)dx, ()
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. ZG participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
Author details
1School of Economics and Trade, Guangdong University of Foreign Studies, Guangzhou, Guangdong 510006, P.R. China.
2Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China.
Acknowledgements
The authors wish to express their thanks to the referees for their careful reading of the manuscript and for their valuable suggestions. This work is supported by the National Natural Science Foundation (No. 61370186), and 2013 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University
(No. 2013KJCX0140).
Received: 20 March 2015 Accepted: 26 September 2015 References
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