Two kinds of Hilbert type integral inequalities in the whole plane

(1)

R E S E A R C H

Open Access

Two kinds of Hilbert-type integral inequalities

in the whole plane

Bicheng Yang

1*

_{and Qiang Chen}

2

*_{Correspondence:}

bcyang@gdei.edu.cn; bcyang818@163.com

1_{Department of Mathematics,}

Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China

Full list of author information is available at the end of the article

Abstract

By the way of using real analysis and estimating the weight functions, two kinds of Hilbert-type integral inequalities in the whole plane with a non-homogeneous kernel and a homogeneous kernel are given. The constant factor related to the triangle functions is proved to be the best possible. We also consider the equivalent forms, the reverses, some particular cases, and two kinds of operator expressions.

MSC: 26D15

Keywords: Hilbert-type integral inequality; weight function; equivalent form; reverse; operator expression

1 Introduction

Assuming thatf(x),g(y)≥,  <_∞f_(x)_dx_<_∞_{and  <}∞

 g(y)dy<∞, we have the

following Hilbert integral inequality (cf.[]):

∞



∞



f(x)g(y)

x+y dx dy<π

∞



f(x)dx

∞



g(y)dy 



, ()

where the constant factorπis the best possible. Inequality () together with the discrete form is important in analysis and its applications (cf.[, ]). In recent years, applying weight functions and introducing parameters, many 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 negative num-ber degrees was given by []. Recently, some inequalities with the homogeneous kernels of degree  and non-homogeneous kernels were studied (cf.[–]). The other kinds of Hilbert-type inequalities were provided by [–]. All of the above integral inequalities are built in the quarter plane of the ﬁrst quadrant.

In , Yang [] ﬁrst gave a Hilbert-type integral inequality in the whole plane as follows:

∞

–∞

∞

–∞

f(x)g(y) ( +ex+y₎λdx dy

<B

λ ,

λ 

∞

–∞e

–λx_f_(x)_dx

∞

–∞e

–λy_g_(y)dy

 

, ()

(2)

where the constant factorB(λ_,λ_) (λ> ) is the best possible, andB(u,v) (u,v> ) is the beta function (cf.[]). Heet al.[–] also provided some Hilbert-type integral inequalities in the whole plane.

In this paper, by the way of using real analysis and estimating the weight functions, two kinds of Hilbert-type integral inequalities in the whole plane with a non-homogeneous kernel and a homogeneous kernel are given. The constant factor related to the triangle functions is proved to be the best possible. We also consider the equivalent forms, the reverses, some particular cases, and two kinds of operator expressions.

2 Some lemmas

In the following, we use the following formula (cf.[]):

∞



(lnt)ta–

t–  dt=

B( –a,a)=

π

sinaπ



( <a< ). ()

Lemma  If <α≤α<π,μ,σ> ,μ+σ=λ,γ ∈ {a;a=__k_+, k–  (k∈N={, , . . .})},

δ∈ {–, },we deﬁne two weight functionsω(σ,y)and (σ,x) (y,x∈R= (–∞,∞))as fol-lows:

ω(σ,y) :=

∞

–∞ max

i∈{,}

ln[|xδ_y|γ_{+ (x}δ_y)γ_cos_α

i] [|xδy|γ _{+ (x}δ_y)γ_cos_α

i]λ/γ–  |y|σ

|x|–δσ dx, ()

(σ,x) :=

_∞

–∞ max

i∈{,}

ln[|xδ_y|γ _{+ (x}δ_y)γ_cos_α

i] [|xδ_y|γ_{+ (x}δ_y)γ_cos_α

i]λ/γ–  |x|δσ

|y|–σ dy. ()

Then for y,x∈R\{},we have

ω(σ,y) = (σ,x) =K(σ),

K(σ) := γ σγ

secα

 σ

γ

+

cscα

 σ

γ _π

λsin(π σ_λ )



∈R₊. ()

Proof Settingu=xδ_y_{in (), for}_y_∈_R_\{__}_{, we ﬁnd}_x₌_y–_δ_u_δ_,_dx₌

δy

–

δuδ–du, and

ω(σ,y) = δ

_–_∞∞max

i∈{,}

ln(|u|γ ₊_uγ_cos_α

i) (|u|γ₊_uγ_cos_α

i)λ/γ – 

|u|σ–_du

=

∞

 max

i∈{,}

ln[uγ_{( +}_cos_α

i)] uλ_{( +}_cos_α

i)λ/γ –  uσ–du

+



–∞ max

i∈{,}

ln[(–u)γ_{( –}_cos_α

i)] (–u)λ_{( –}_cos_α

i)λ/γ– 

(–u)σ–_du

=γ

∞

 max

i∈{,}

ln[u( +cosαi)/γ] [u( +cosαi)/γ]λ– 

uσ–du

+

∞

 max

i∈{,}

ln[u( –cosαi)/γ] [u( –cosαi)/γ]λ– 

uσ–_du

(3)

In view of the function_tlnλ_–t being strictly decreasing in R+(cf.[]), we have

ω(σ,y) =

∞



)]

[u( +cosα)/γ]λ– 

uσ–du

+

∞



ln[uγ_{( –}_cos_α

)]

[u( –cosα)/γ]λ– 

uσ–_du. _()

Settingt= [u( +cosα)/γ]λ(t= [u( –cosα)/γ]λ) in the above ﬁrst (second) integral, by

calculations and (), it follows that

ω(σ,y) = γ λ

_secα



 σ

γ

+

_cscα



σ

γ ∞

 lnt t– t

σ

λ–_dt₌_K(_σ_).

Settingu=xδ_y_{in (), for}_x_∈_R_\{__}_{, we ﬁnd}_y₌_x–δ_u,_dy₌_x–δ_du_and

(σ,x) =

_∞

–∞ max

i∈{,}

ln(|u|γ₊_uγ_cos_α

i)|u|σ– (|u|γ₊_uγ_cos_α

i)λ/γ – 

du=K(σ).

Hence we have ().

Remark  If we replacemax_i_∈{_,_}bymin_i_∈{_,_}in () and (), then we must exchangeα

andαin ().

Lemma  If p> ,

p+



q= ,  <α≤α<π,μ,σ> ,μ+σ=λ,γ ∈ {a;a=



k+, k–  (k∈

N)},δ∈ {–, },K(σ)is indicated by(),f(x)is a non-negative measurable function inR, then we have

J:=

∞

–∞|y|

pσ–

∞

–∞ max

i∈{,}

i]f(x) [|xδ_y_|γ_{+ (x}δ_y)γ_cos_α

i]λ/γ–  dx

p dy

≤Kp(σ)

_∞

–∞|x|

p(–δσ)–_fp_(x)_dx. _()

Proof For simplifying in the following, we set

h(δ)_(x,_{y) :=}_max

i∈{,}

ln[|xδ_y_|γ _{+ (x}δ_y)γ_cos_αi_]

[|xδy|γ_{+ (x}δ_y)γ_cos_α

i]λ/γ– 

(x,y∈R). ()

By Hölder’s inequality (cf.[]), we have

∞

–∞h

(δ)_(x,_y)f_(x)_dx

p

=

∞

–∞

h(δ)(x,y)

|x|(–δσ)/q |y|(–σ)/p f(x)

|y|(–σ)/p |x|(–δσ)/q dx

p

≤

∞

–∞

h(δ)(x,y)|x|

(–δσ)(p–)

|y|–σ f

p_(x)_dx

∞

–∞

h(δ)(x,y)|y|

(–σ)(q–)

|x|–δσ dx

p–

=ω(σ,y)p–|y|–pσ+

_∞

–∞h

(δ)_(x,_y)|x|(–δσ)(p–)

|y|–σ f

(4)

Then by () and the Fubini theorem (cf.[]), it follows that

J≤Kp–(σ)

∞

–∞

∞

–∞h

(δ)_(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  If p > , _p + _q = ,  < α ≤α < π, μ,σ > , μ+σ =λ, γ ∈ {a;a= 

k+, k–  (k∈N)}, δ∈ {–, }, K(σ) is indicated by(), f(x),g(y)≥,satisfying <

∞

–∞|x|

p(–δσ)–_fp_(x)_dx_<_∞_and_{ <}∞

–∞|y|

q(–σ)–_gq_(y)_dy_<_∞,_{then we have}

I:=

∞

–∞

∞

–∞ max

i∈{,}

ln[|xδ_y_|γ_{+ (x}δ_y)γ_cos_α

i] [|xδy|γ_{+ (x}δ_y)γ_cos_α

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–

∞

–∞ max

i∈{,}

ln[|xδ_y_|γ_{+ (x}δ_y)γ_cos_α

i] [|xδy|γ_{+ (x}δ_y)γ_cos_α

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.}_Inequalities_()_and_() are equivalent.

In particular, forα=α=α∈(,π),γ=  in () and (), we ﬁnd

K(σ) =k(σ) :=  σ

secα

 σ

+

cscα

 σ

π λsin(π σ

λ )



, ()

and the following equivalent inequalities:

_∞

–∞

_∞

–∞

ln(|xδ_y|₊_xδ_y_cos_α₎

(|xδ_y_|₊_xδ_y_cos_α₎λ_{– }f(x)g(y)dx dy

<k(σ)

∞

–∞|x|

p(–δσ)–_fp_(x)_dx

 p ∞

–∞|y|

q(–σ)–_gq_(y)_dy

 q

, ()

_∞

–∞|y|

pσ–

∞

–∞

ln(|xδ_y|₊_xδ_y_cos_α₎

(|xδ_y|₊_xδ_y_cos_α₎λ_{– }f(x)dx

p dy

<kp(σ)

∞

–∞|x|

p(–σ)–_fp_(x)dx. _()

Proof If () takes the form of an equality for ay= , then there exist constantsAandB, such that they are not all zero, and

A|x|

(–δσ)(p–)

|y|–σ f

p_{(x) =}_B|y|(–σ)(q–)

(5)

We suppose thatA=  (otherwiseB=A= ). Then it follows that

|x|p(–δσ)–_fp_{(x) =}_|y|q(–σ) B

A|x| a.e. in R,

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 the Hölder inequality (cf.[]), we ﬁnd

I =

∞

–∞

|y|σ–p

∞

–∞h

(δ)_(x,_y)f_(x)_dx_|_y_|p–σ_g(y)_dy

≤Jp

∞

–∞

|y|q(–σ)–_gq_(y)_dy

 q

. ()

Then by (), we have (). On the other hand, suppose that () is valid. Setting

g(y) :=|y|pσ–

∞

–∞h

(δ)_(x,_y)f_(x)_dx

p–

(y∈R),

then it follows thatJ=_–∞_∞|y|q(–σ)–_gq_(y)_{dy. By (), we have}_J_<_{∞. If}_J_{= , then () is} trivially true; 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

<∞, ()

Jp₌

∞

–∞

|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≥}, andEδ+:=Eδ∩R+={x∈R+;xδ≥}. Forε> , we deﬁne

two functionsf˜(x),g(y) as follows:˜

˜ f(x) :=

|x|δ(σ–pε)–_, _x∈_E

δ,

, 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

= 

E_δ+

x–δε–_dx



p 



yε–_dy

 q

= ε.

We ﬁnd

I(x) :=



– max

i∈{,}

ln[|xδ_y_|γ_{+ (x}δ_y)γ_cos_αi_]_|_y_|σ+_qε–

[|xδy|γ_{+ (x}δ_y)γ_cos_α

i]λ/γ– 

(6)

and thenI(x) is an even function. In fact, settingY= –y, we obtain

I(–x) =



– max

i∈{,}

ln[|(–x)δ_y_|γ_{+ ((–x)}δ_y)γ_cos_α

i]|y|

σ+_qε–

[|(–x)δy|γ_{+ ((–x)}δ_y)γ_cos_α

i]λ/γ –  dy

=



– max

i∈{,}

ln[|xδ_Y|γ _{+ (x}δ_Y)γ_cos_α

i]|Y|σ+

ε

q– [|xδY|γ_{+ (x}δ_Y)γ_cos_α

i]λ/γ– 

dY=I(x).

It follows that

˜ I =

_∞

–∞

_∞

–∞h

(δ)_(x,_y)_f˜_(x)˜_g(y)_{dx dy}

=

Eδ

|x|δ(σ–_pε)–

I(x)dx= 

E+_δ

xδ(σ–pε)–_I(x)_dx

u=xδ_y

= 

E+_δ x–δε–

xδ

–xδ max

i∈{,}

i)|u|σ+

ε

q– (|u|γ₊_uγ_cos_α

i)λ/γ– 

du dx.

Settingv=xδ _{in the above integral, by the Fubini theorem (cf.}_{[]), we ﬁnd}

˜ I= 

_∞



v–ε–

v

–v

max

i∈{,}

i)|u|σ+

ε

q–

(|u|γ₊_uγ_cos_α

i)λ/γ– 

du dv

= 

∞



v–ε–

v



max

i∈{,}

i)] [u( +cosαi)/γ]λ– 

+max

i∈{,}

i)] [u( –cosαi)/γ]λ– 

uσ+qε–_du

dv

= 

∞



v–ε–

v



)]

[u( +cosα)/γ]λ– 

+ ln[u

γ_{( –}_cos_α

)]

[u( –cosα)/γ]λ– 

dv

= 

∞



v–ε–





)]

[u( +cosα)/γ]λ– 

+ ln[u

γ_{( –}_cos_α

)]

[u( –cosα)/γ]λ– 

dv

+ 

∞



v–ε–

v



)]

[u( +cosα)/γ]λ– 

+ ln[u

γ_{( –}_cos_α

)]

[u( –cosα)/γ]λ– 

uσ+qε–_du dv = ε  

)]u

σ+_qε–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]u

σ+_qε–

[u( –cosα)/γ]λ– 

du +  ∞  ∞ u

v–ε–_dv

×

)]uσ+

ε

q–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ+

ε

q–

[u( –cosα)/γ]λ– 

du = ε  

)]u

σ+_qε–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]u

σ+_qε–

[u( –cosα)/γ]λ– 

du

+

∞



)]uσ–

ε

p–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ–

ε

p–

[u( –cosα)/γ]λ– 

du

(7)

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˜





_ln

[uγ_{( +}_cos_α

)]uσ+

ε

q–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ+

ε

q–

[u( –cosα)/γ]λ– 

du

+

∞



)]uσ–

ε

p–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ–

ε

p–

[u( –cosα)/γ]λ– 

du

=ε˜I<εkL˜=k. ()

By () and the Levi theorem (cf.[]), we have

K(σ) =

∞



)]uσ–du

[u( +cosα)/γ]λ– 

+

∞



)]uσ–du

[u( –cosα)/γ]λ– 

=



 lim

ε→+

)]uσ+

ε

q–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ+

ε

q–

[u( –cosα)/γ]λ– 

du

+

∞

 lim

ε→+

)]uσ–

ε

p–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ–

ε

p–

[u( –cosα)/γ]λ– 

du

= lim

ε→+





)]uσ+

ε

q–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ+

ε

q–

[u( –cosα)/γ]λ– 

du

+

∞



)]uσ–

ε

p–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ–

ε

p–

[u( –cosα)/γ]λ– 

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 would reach a

contradiction: that the constant factor in () is not the best possible.

Theorem  On 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 inequality (cf.[]), we have the reverses of () and (). It is easy to obtain the reverse of (). In view of the reverses of () and (), we obtain the reverse of (). On the other hand, suppose that the reverse of () is valid. Setting the same g(y) as Theorem , by the reverse of (), we haveJ> . IfJ=∞, then the reverse of () is trivially value; ifJ<∞, then by the reverse of (), we obtain the reverses 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





)]u

σ+_qε–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]u

σ+_qε–

[u( –cosα)/γ]λ– 

du

+

∞



)]uσ–

ε

p–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ–

ε

p–

[u( –cosα)/γ]λ– 

(8)

Forε→+_{, by the Levi theorem (cf.}_{[]), we ﬁnd}

∞



)]u

σ–_pε–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]u

σ–_pε–

[u( –cosα)/γ]λ– 

du

→ ∞



)]uσ–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ–

[u( –cosα)/γ]λ– 

du. ()

There exists a constantδ> , such thatσ–_δ> , and thenK(σ–δ_)∈R+. For  <

ε<δ|q|

 (q< ), sinceu

σ+_qε–_≤

uσ–δ–_,_u∈_{(, ], and}

 <





)]uσ– δ

– [u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ– δ

– [u( –cosα)/γ]λ– 

du

≤K

σ–δ



,

then by the Lebesgue control convergence theorem (cf.[]), forε→+_{, we have}





)]uσ+

ε

q–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ+

ε

q–

[u( –cosα)/γ]λ– 

du

→





)]uσ–

[u( +cosα)/γ]λ– 

+ln[u

γ_{( –}_cos_α

)]uσ–

[u( –cosα)/γ]λ– 

du. ()

By (), (), and (), forε→+_{, we ﬁnd}_K(_σ₎_≥_{k, which contradicts the fact that}_k_>

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 would reach a contradiction: that the constant factor in the reverse of () is not

the best possible.

Corollary  Forδ= –in()and(),replacing|x|λ_f_(x)_{by f}_(x),_{we obtain}

 <

∞

–∞|x|

p(–μ)–_fp_(x)_dx_<_∞,

and the following equivalent inequalities with the homogeneous kernel and the best possible constant factors:

∞

–∞

∞

–∞ max

i∈{,}

ln[(|y|γ ₊_sgn_(x)yγ_cos_αi_)/_|_x_|γ_]

[|y|γ ₊_sgn_(x)yγ_cos_α

i]λ/γ–|x|λ

f(x)g(y)dx dy

<K(σ)

∞

–∞

|x|p(–μ)–_fp_(x)_dx

 p ∞

–∞

|y|q(–σ)–_gq_(y)_dy

 q

, ()

∞

–∞|y|

pσ–

∞

–∞ max

i∈{,}

ln[(|y|γ₊_sgn_(x)yγ_cos_α

i)/|x|γ] [|y|γ₊_sgn_(x)yγ_cos_α

i]λ/γ –|x|λ 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

(9)

∞

–∞

∞

–∞

ln[(|y|+sgn(x)ycosα)/|x|]

[|y|+sgn(x)ycosα]λ_–|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–

∞

–∞

ln[(|y|+sgn(x)ycosα)/|x|] [|y|+sgn(x)ycosα]λ_–|x|λf(x)dx

p dy

<kp(σ)

∞

–∞

|x|p(–μ)–_fp_(x)_dx, _()

wherek(σ) is indicated by ().

4 Two kinds of operator expressions

Suppose thatp> ,_p+_q= ,  <α≤α<π,μ,σ> ,μ+σ=λ,γ∈ {a;a=__k_+, k–  (k∈

N)},δ∈ {–, }. We set the following functions:

ϕ(x) :=|x|p(–δσ)–_, _ψ_{(y) :=}_|_y_|q(–σ)–_, _φ_{(x) :=}_|_x_|p(–μ)– _(x,_y_∈_R_),

therefore,ψ–p(y) =|y|pσ–_{. Deﬁne the following real normed linear space:}

Lp,ϕ(R) :=

f:fp,ϕ:=

∞

–∞ϕ(x)

f(x)pdx 

p <∞

,

Lp,ψ–p(R) =

h:hp,ψ–p=

∞

–∞

ψ–p(y)h(y)pdy 

p <∞

,

Lp,φ(R) =

g:gp,φ=

∞

–∞

φ(x)g(x)pdx 

p <∞

.

(a) In view of Theorem , forf ∈Lp,ϕ(R), setting

H(y) :=

∞

–∞ max

i∈{,}

i]

[|xδ_y_|γ _{+ (x}δ_y)γ_cos_αi_]λ/γ_{– }f(x)dx (y∈R),

by (), we have

Hp,ψ–p=

∞

–∞ψ

–p_(y)Hp

(y)dy

 p

<K(σ)fp,ϕ<∞. ()

Deﬁnition  Deﬁne a Hilbert-type integral operator with the non-homogeneous kernel in the whole planeT:Lp,ϕ(R)→Lp,ψ–p(R) as follows: For anyf ∈Lp,ϕ(R), there exists a

unique representationTf =H∈Lp,ψ–p(R), satisfying, for anyy∈R,Tf(y) =H(y).

In view of (), it follows thatTfp,ψ–p=Hp,ψ–p≤K(σ)fp,ϕ, and then the

oper-atorTis bounded satisfying

T= sup

f(=θ)∈Lp,ϕ(R)

Tfp,ψ–p fp,ϕ

≤K(σ).

(10)

If we deﬁne the formal inner product ofTf andgas

(Tf,g) :=

∞

–∞

∞

–∞h

(δ)_(x,_y)f_(x)_dx

g(y)dy

=

_∞

–∞

_∞

–∞h

(δ)_(x,_y)f_(x)g(y)_{dx dy,}

then we can rewrite () and () as follows:

(Tf,g) <T · fp,ϕgq,ψ, Tfp,ψ–p<T_ · f_p_,ϕ.

(b) In view of Corollary , forf ∈Lp,φ(R), setting

H(y) :=

∞

–∞ max

i∈{,}

ln[(|y|γ₊_sgn_(x)yγ_cos_α

i)/|x|γ] [|y|γ ₊_sgn_(x)yγ_cos_α

i]λ/γ–|x|λ

f(x)dx (y∈R),

by (), we have

Hp,ψ–p=

∞

–∞

ψ–p(y)H_p(y)dy 

p

<K(σ)fp,φ<∞. ()

Deﬁnition  Deﬁne a Hilbert-type integral operator with the homogeneous kernel in the whole planeT:Lp,φ(R)→Lp,ψ–p(R) as follows: For anyf∈L_p_,φ(R), there exists a unique

representationTf =H∈Lp,ψ–p(R), satisfying, for anyy∈R,T_f(y) =H_(y).

In view of (), it follows thatTfp,ψ–p=H__p_,_ψ–p≤K(σ)f_p_,φ, and then the

oper-atorTis bounded satisfying

T= sup

f(=θ)∈Lp,φ(R)

Tfp,ψ–p fp,φ ≤

K(σ).

Since the constant factorK(σ) in () is the best possible, we haveT=K(σ).

If we deﬁne the formal inner product ofTf andgas

(Tf,g) :=

∞

–∞

∞

–∞ max

i∈{,}

ln[(|y|γ ₊_sgn_(x)yγ_cos_α

i)/|x|γ] [|y|γ₊_sgn_(x)yγ_cos_α

i]λ/γ–|x|λ

f(x)g(y)dx dy,

then we can rewrite () and () as follows:

(Tf,g) <T · fp,φgq,ψ, Tfp,ψ–p<T_ · f_p_,φ.

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. QC participated in the design of the study and performed the numerical analysis. All authors read and approved the ﬁnal manuscript.

Author details

(11)

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 61370186), and 2013 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2013KJCX0140).

Received: 23 August 2014 Accepted: 24 December 2014

References

1. Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1934)

2. Mitrinovi´c, DS, Pecaric, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991)

3. Yang, BC: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009)

4. Yang, BC: A survey of the study of Hilbert-type inequalities with parameters. Adv. Math.38(3), 257-268 (2009) 5. Yang, BC: On the norm of an integral operator and applications. J. Math. Anal. Appl.321, 182-192 (2006) 6. Xu, JS: Hardy-Hilbert’s inequalities with two parameters. Adv. Math.36(2), 63-76 (2007)

7. Yang, BC: On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl.325, 529-541 (2007) 8. Xin, DM: A Hilbert-type integral inequality with the homogeneous kernel of zero degree. Math. Theory Appl.30(2),

70-74 (2010)

9. Yang, BC: A Hilbert-type integral inequality with the homogeneous kernel of degree 0. J. Shandong Univ. Nat. Sci. 45(2), 103-106 (2010)

10. Debnath, L, Yang, BC: Recent developments of Hilbert-type discrete and integral inequalities with applications. Int. J. Math. Math. Sci.2012, Article ID 871845 (2012)

11. Rassias, MT, Yang, BC: On half-discrete Hilbert’s inequality. Appl. Math. Comput.220, 75-93 (2013)

12. Yang, BC, Krnic, M: A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree 0. J. Math. Inequal.6(3), 401-417 (2012)

13. Rassias, MT, Yang, BC: A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function. Appl. Math. Comput.225, 263-277 (2013)

14. Rassias, MT, Yang, BC: On a multidimensional half - discrete Hilbert - type inequality related to the hyperbolic cotangent function. Appl. Math. Comput.242, 800-813 (2013)

15. Rassias, MT, Yang, BC: A multidimensional Hilbert-type integral inequality related to the Riemann zeta function. In: Daras, NJ (ed.) Applications of Mathematics and Informatics in Science and Engineering, pp. 417-433. Springer, New York (2014)

16. Yang, BC: A new Hilbert-type integral inequality. Soochow J. Math.33(4), 849-859 (2007) 17. Wang, ZQ, Guo, DR: Introduction to Special Functions. Science Press, Beijing (1979)

18. He, B, Yang, BC: On a Hilbert-type integral inequality with the homogeneous kernel of 0-degree and the hypergeometric function. Math. Pract. Theory40(18), 105-211 (2010)

19. Yang, BC: A new Hilbert-type integral inequality with some parameters. J. Jilin Univ. Sci. Ed.46(6), 1085-1090 (2008) 20. Yang, BC: A Hilbert-type integral inequality with a non-homogeneous kernel. J. Xiamen Univ. Natur. Sci.48(2),

165-169 (2008)

21. Zeng, Z, Xie, ZT: On a new Hilbert-type integral inequality with the homogeneous kernel of degree 0 and the integral in whole plane. J. Inequal. Appl.2010, Article ID 256796 (2010)

22. Yang, BC: A reverse Hilbert-type integral inequality with some parameters. J. Xinxiang Univ. Natur. Sci. Ed.27(6), 1-4 (2010)

23. Wang, AZ, Yang, BC: A new Hilbert-type integral inequality in whole plane with the non-homogeneous kernel. J. Inequal. Appl.2011, Article ID 123 (2011). doi:10.1186/1029-24X-2011-123

24. Xin, DM, Yang, BC: A Hilbert-type integral inequality in whole plane with the homogeneous kernel of degree –2. J. Inequal. Appl.2011, Article ID 401428 (2011)

25. He, B, Yang, BC: On an inequality concerning a non-homogeneous kernel and the hypergeometric function. Tamsui Oxf. J. Inf. Math. Sci.27(1), 75-88 (2011)

26. Yang, BC: A reverse Hilbert-type integral inequality with a non-homogeneous kernel. J. Jilin Univ. Sci. Ed.49(3), 437-441 (2011)

27. Xie, ZT, Zeng, Z, Sun, YF: A new Hilbert-type inequality with the homogeneous kernel of degree –2. Adv. Appl. Math. Sci.12(7), 391-401 (2013)

28. Zeng, Z, Raja Rama Gandhi, K, Xie, ZT: A new Hilbert-type inequality with the homogeneous kernel of degree –2 and with the integral. Bull. Math. Sci. Appl.3(1), 11-20 (2014)