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On a generalization of a Hilbert-type integral inequality in the whole plane with a hypergeometric function

Abstract

By introducing some parameters, we establish a generalization of the Hilbert-type integral inequality in the whole plane with the homogeneous kernel of degree 2λ and the best constant factor which involves the hypergeometric function.

MSC:26D15.

1 Introduction

If p>1, 1 p + 1 q =1 and f(x),g(x)0 satisfy

0< 0 f p (x)dx<and0< 0 g q (x)dx<,

then

0 0 f ( x ) g ( y ) x + y dxdy< π sin ( π / p ) { 0 f p ( x ) d x } 1 p { 0 g q ( x ) d x } 1 q ,
(1)

where the constant factor π/(sinπ/p) is the best possible. Inequality (1) is called Hardy-Hilbert’s inequality [1] and is important in analysis and applications [2].

In 2001, Yang gave an extension of (1) involving beta function as (see [3]):

0 0 f ( x ) g ( y ) ( x + y ) λ d x d y < B ( p + λ 2 p , q + λ 2 q ) { 0 x 1 λ f p ( x ) d x } 1 p { 0 x 1 λ g q ( x ) d x } 1 q ,
(2)

where the constant factor B( p + λ 2 p , q + λ 2 q ) (λ>2min{p,q}) is the best possible.

Recently, some new Hilbert-type inequalities in the whole plane have been obtained [4, 5]. Xin and Yang in [5] established the following:

If p>1, 1 p + 1 q =1, |β|<1, 0< α 1 < α 2 <π, f,g0, satisfy

0< | x | p β 1 f p (x)dx<and0< | y | q β 1 g q (y)dy<,

then we have

min i { 1 , 2 } { 1 x 2 + 2 x y cos α i + y 2 } f ( x ) g ( y ) d x d y < k ( β ) ( | x | p β 1 f p ( x ) d x ) 1 p ( | y | q β 1 g q ( y ) d y ) 1 q ,
(3)

and

| y | p ( 1 β ) 1 ( min i { 1 , 2 } { 1 x 2 + 2 x y cos α i + y 2 } f ( x ) d x ) p d y < k p ( β ) | x | p β 1 f p ( x ) d x ,
(4)

where the constant factors k(β)= π sin β π ( sin β α 1 sin α 1 + sin β ( π α 2 ) sin α 2 ) and k p (β) are the best possible. Inequalities (3) and (4) are equivalent.

By introducing some parameters, we establish generalizations of inequalities (3) and (4) with the homogeneous kernel of degree 2λ and the best constant factor which involves the hypergeometric function.

2 Preliminary lemmas

In order to prove our assertions, we need the following lemmas.

Recall that the hypergeometric function F(α,β;γ;x) is defined [6] by

F(α,β;γ;x)= r = 0 ( α ) r ( β ) r ( γ ) r x r r ! ,
(5)

where ( α ) r is the Pochhammer symbol defined by

( α ) r =α(α+1)(α+r1)= Γ ( α + r ) Γ ( α ) .

It is known the series (5) converges for |x|<1 and diverges for |x|>1. The hypergeometric function satisfies the integral representation

F(α,β;γ;x)= Γ ( γ ) Γ ( β ) Γ ( γ β ) 0 1 t β 1 ( 1 t ) γ β 1 ( 1 x t ) α dt,if γ>β>0.

Lemma 2.1 (See [7])

Suppose that a,c>0, b 2 <ac, 0<α<2λ. Then we have

0 x α 1 ( a x 2 + 2 b x + c ) λ dx= a α 2 c α 2 λ B(α,2λα)F ( α 2 , λ α 2 ; λ + 1 2 ; 1 b 2 a c ) .

Lemma 2.2 Let a,c,λ>0, b0, 12λ<β<1 and 0< α 1 < α 2 <π be real parameters such that b 2 max{ cos 2 α 1 , cos 2 (π α 2 )}<ac. Define the weight functions ω(x) and ϖ(y) (x,y(,)) as follows:

ω ( x ) : = min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } | x | β + 2 λ 1 | y | β d y , ϖ ( y ) : = min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } | y | 1 β | x | β 2 λ + 2 d x .

Then we have ω(x)=ϖ(y)= C λ (x,y0), where

C λ = a 1 β 2 λ c 1 β 2 B ( 1 β , 2 λ + β 1 ) × [ F ( 1 β 2 , λ 1 β 2 ; λ + 1 2 ; 1 b 2 cos 2 α 1 a c ) + F ( 1 β 2 , λ 1 β 2 ; λ + 1 2 ; 1 b 2 cos 2 ( π α 2 ) a c ) ] .

Proof For x(,0), setting u=y/x, u=y/x in the following two integrals, respectively, and using Lemma 2.1, we get

ω ( x ) = 0 1 ( a x 2 + 2 b x y cos α 1 + c y 2 ) λ ( x ) β + 2 λ 1 ( y ) β d y + 0 1 ( a x 2 + 2 b x y cos α 2 + c y 2 ) λ ( x ) β + 2 λ 1 y β d y = 0 u β ( c u 2 + 2 b u cos α 1 + a ) λ d u + 0 u β ( c u 2 + 2 b u cos ( π α 2 ) + a ) λ d u = C λ .

For x(0,), setting u=y/x, u=y/x in the following two integrals, respectively, and using Lemma 2.1, we get

ω ( x ) = 0 1 ( a x 2 + 2 b x y cos α 2 + c y 2 ) λ x β + 2 λ 1 ( y ) β d y + 0 1 ( a x 2 + 2 b x y cos α 1 + c y 2 ) λ x β + 2 λ 1 y β d y = 0 u β ( c u 2 + 2 b u cos ( π α 2 ) + a ) λ d u + 0 u β ( c u 2 + 2 b u cos α 1 + a ) λ d u = C λ .

By the same way, we still can find that ω(x)=ϖ(y)= C λ (x,y0). The lemma is proved. □

Lemma 2.3 Let p and q be conjugate parameters with p>1, and let a,c,λ>0, b0, 12λ<β<1, 0< α 1 < α 2 <π and b 2 max{ cos 2 α 1 , cos 2 (π α 2 )}<ac, and f(x) be a nonnegative measurable function in (,), then we have

J : = | y | p ( 1 β ) 1 ( min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } f ( x ) d x ) p d y C λ p | x | p ( β + 2 λ 2 ) 1 f p ( x ) d x .
(6)

Proof By Lemma 2.2 and Hölder’s inequality [8], we have

( min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } f ( x ) d x ) p = [ min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } × ( | x | ( β 2 λ + 2 ) / q | y | β / p f ( x ) ) ( | y | β / p | x | ( β 2 λ + 2 ) / q ) d x ] p min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } | x | ( 1 p ) ( β + 2 λ 2 ) | y | β f p ( x ) d x × ( min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } | y | ( q 1 ) β | x | ( β 2 λ + 2 ) d x ) p 1 = C λ p 1 | y | p ( β 1 ) + 1 min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } | x | ( 1 p ) ( β + 2 λ 2 ) | y | β f p ( x ) d x .
(7)

Then by the Fubini theorem, it follows that

J C λ p 1 [ min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } | x | ( 1 p ) ( β + 2 λ 2 ) | y | β f p ( x ) d x ] d y = C λ p 1 ω ( x ) | x | p ( β + 2 λ 2 ) 1 f p ( x ) d x = C λ p | x | p ( β + 2 λ 2 ) 1 f p ( x ) d x .

The lemma is proved. □

3 Main results

Theorem 3.1 Let p and q be conjugate parameters with p>1, and let a,c,λ>0, b0, 12λ<β<1, 0< α 1 < α 2 <π and b 2 max{ cos 2 α 1 , cos 2 (π α 2 )}<ac, and f,g0, satisfy 0< | x | p ( β + 2 λ 2 ) 1 f p (x)dx< and 0< | y | q β 1 g q (y)dy<. Then

I : = min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } f ( x ) g ( y ) d x d y < C λ ( | x | p ( β + 2 λ 2 ) 1 f p ( x ) d x ) 1 / p ( | y | q β 1 g q ( y ) d y ) 1 / q ,
(8)

and

J = | y | p ( 1 β ) 1 ( min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } f ( x ) d x ) p d y < C λ p | x | p ( β + 2 λ 2 ) 1 f p ( x ) d x ,
(9)

where the constant factors C λ and C λ p are the best possible and C λ is defined in Lemma  2.2. Inequalities (8) and (9) are equivalent.

Proof If (7) takes the form of the equality for a y(,0)(0,), then there exist constants A and B such that they are not all zero, and

A | x | ( 1 p ) ( β + 2 λ 2 ) | y | β f p (x)=B | y | ( q 1 ) β | x | ( β 2 λ + 2 ) a.e. in (,)×(,).

Hence, there exists a constant K such that

A | x | p ( β + 2 λ 2 ) f p (x)=B | y | q β =Ka.e. in (,)×(,).

We suppose A0 (otherwise B=A=0). Then | x | p ( β + 2 λ 2 ) 1 f p (x)=K/(A|x|) a.e. in (,), which contradicts the fact that 0< | x | p ( β + 2 λ 2 ) 1 f p (x)dx<. Hence, (7) takes the form of a strict inequality, so does (6), and we have (9).

By Hölder’s inequality [8], we have

I = ( | y | 1 / q β min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } f ( x ) d x ) × ( | y | β 1 / q g ( y ) ) d y J 1 / p ( | y | q β 1 g q ( y ) d y ) 1 / q .
(10)

By (9), we have (8). On the other hand, suppose that (8) is valid. Set

g(y)= | y | p ( 1 β ) 1 ( min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } f ( x ) d x ) p 1 ,

then it follows J= | y | q β 1 g q (y)dy. By (6), we have J<. If J=0, then (9) is obviously valid. If 0<J<, then by (8), we obtain

0 < | y | q β 1 g q ( y ) d y = J = I < C λ ( | x | p ( β + 2 λ 2 ) 1 f p ( x ) d x ) 1 / p ( | y | q β 1 g q ( y ) d y ) 1 / q ,

and

J 1 / p = ( | y | q β 1 g q ( y ) d y ) 1 / p < C λ ( | x | p ( β + 2 λ 2 ) 1 f p ( x ) d x ) 1 / p .

Hence, we have (9), which is equivalent to (8).

For ε>0, define functions f ˜ (x), g ˜ (y) as follows:

f ˜ ( x ) : = { x ( β + 2 λ 2 ) 2 ε / p , x ( 1 , ) , 0 , x [ 1 , 1 ] , ( x ) ( β + 2 λ 2 ) 2 ε / p , x ( , 1 ) , g ˜ ( y ) : = { y β 2 ε / q , y ( 1 , ) , 0 , y [ 1 , 1 ] , ( y ) β 2 ε / q , y ( , 1 ) .

Then

L ˜ := ( | x | p ( β + 2 λ 2 ) 1 f ˜ p ( x ) d x ) 1 / p ( | y | q β 1 g ˜ q ( y ) d y ) 1 / q = 1 ε ,

and

I ˜ : = min i { 1 , 2 } { 1 ( a x 2 + 2 b x y cos α i + c y 2 ) λ } f ˜ ( x ) g ˜ ( y ) d x d y = I 1 + I 2 + I 3 + I 4 ,

where

I 1 : = 1 ( x ) ( β + 2 λ 2 ) 2 ε / p [ 1 ( y ) β 2 ε / q ( a x 2 + 2 b x y cos α 1 + c y 2 ) λ d y ] d x , I 2 : = 1 ( x ) ( β + 2 λ 2 ) 2 ε / p [ 1 y β 2 ε / q ( a x 2 + 2 b x y cos α 2 + c y 2 ) λ d y ] d x , I 3 : = 1 x ( β + 2 λ 2 ) 2 ε / p [ 1 ( y ) β 2 ε / q ( a x 2 + 2 b x y cos α 2 + c y 2 ) λ d y ] d x ,

and

I 4 := 1 x ( β + 2 λ 2 ) 2 ε / p [ 1 y β 2 ε / q ( a x 2 + 2 b x y cos α 1 + c y 2 ) λ d y ] dx.

Taking u=y/x, by the Fubini theorem, we obtain

I 1 = I 4 = 1 x 1 2 ε 1 / x u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u d x = 1 x 1 2 ε ( 1 / x 1 u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u + 1 u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u ) d x = 0 1 ( 1 / u x 1 2 ε d x ) u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u + 1 2 ε 1 u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u = 1 2 ε ( 0 1 u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u + 1 u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u ) ,

and

I 2 = I 3 = 1 2 ε ( 0 1 u β 2 ε / q ( c u 2 2 b u cos α 2 + a ) λ d u + 1 u β 2 ε / q ( c u 2 2 b u cos α 2 + a ) λ d u ) .

In view of the above results, if the constant factor C λ in (8) is not the best possible, then there exists a positive number C ˜ with C ˜ < C λ such that

0 1 u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u + 1 u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u + 0 1 u β 2 ε / q ( c u 2 2 b u cos α 2 + a ) λ d u + 1 u β 2 ε / q ( c u 2 2 b u cos α 2 + a ) λ d u = ε I ˜ < ε C ˜ L ˜ = C ˜ .
(11)

By the Fatou lemma and (11), we have

C λ = 0 u β ( c u 2 + 2 b u cos α 1 + a ) λ d u + 0 u β ( c u 2 2 b u cos α 2 + a ) λ d u = 0 1 lim ε 0 + u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u + 1 lim ε 0 + u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u + 0 1 lim ε 0 + u β 2 ε / q ( c u 2 2 b u cos α 2 + a ) λ d u + 1 lim ε 0 + u β 2 ε / q ( c u 2 2 b u cos α 2 + a ) λ d u lim ε 0 + [ 0 1 u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u + 1 u β 2 ε / q ( c u 2 + 2 b u cos α 1 + a ) λ d u + 0 1 u β 2 ε / q ( c u 2 2 b u cos α 2 + a ) λ d u + 1 u β 2 ε / q ( c u 2 2 b u cos α 2 + a ) λ d u ] C ˜ ,

which contradicts the fact that C ˜ < C λ . Hence, the constant factor C λ in (8) is the best possible.

If the constant factor in (9) is not the best possible, then by (10), we may get a contradiction that the constant factor in (8) is not the best possible. Thus the theorem is proved. □

Remark 1 Setting λ=a=b=c=1 in Theorem 3.1, we have (3) and (4).

Remark 2 Setting λ=1/2 in Theorem 3.1, we have the following particular results:

min i { 1 , 2 } { 1 a x 2 + 2 b x y cos α i + c y 2 } f ( x ) g ( y ) d x d y < C 1 / 2 ( | x | p ( β 1 ) 1 f p ( x ) d x ) 1 / p ( | y | q β 1 g q ( y ) d y ) 1 / q ,

and

| y | p ( 1 β ) 1 ( min i { 1 , 2 } { 1 a x 2 + 2 b x y cos α i + c y 2 } f ( x ) d x ) p d y < C 1 / 2 p | x | p ( β 1 ) 1 f p ( x ) d x .

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Correspondence to Tserendorj Batbold.

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Adiyasuren, V., Batbold, T. On a generalization of a Hilbert-type integral inequality in the whole plane with a hypergeometric function. J Inequal Appl 2013, 189 (2013). https://doi.org/10.1186/1029-242X-2013-189

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Keywords

  • Hilbert’s inequality
  • Hölder’s inequality
  • homogeneous kernel
  • weight function
  • equivalent form