Open Access

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

Journal of Inequalities and Applications20132013:189

https://doi.org/10.1186/1029-242X-2013-189

Received: 17 May 2012

Accepted: 16 January 2013

Published: 19 April 2013

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 2 λ and the best constant factor which involves the hypergeometric function.

MSC:26D15.

Keywords

Hilbert’s inequalityHölder’s inequalityhomogeneous kernelweight functionequivalent form

1 Introduction

If p > 1 , 1 p + 1 q = 1 and f ( x ) , g ( x ) 0 satisfy
0 < 0 f p ( x ) d x < and 0 < 0 g q ( x ) d x < ,
then
0 0 f ( x ) g ( y ) x + y d x d y < 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 / ( sin π / 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 > 2 min { 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 , g 0 , satisfy
0 < | x | p β 1 f p ( x ) d x < and 0 < | y | q β 1 g q ( y ) d y < ,
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 2 λ 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 + r 1 ) = 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 d t , if  γ > β > 0 .

Lemma 2.1 (See [7])

Suppose that a , c > 0 , b 2 < a c , 0 < α < 2 λ . Then we have
0 x α 1 ( a x 2 + 2 b x + c ) λ d x = a α 2 c α 2 λ B ( α , 2 λ α ) F ( α 2 , λ α 2 ; λ + 1 2 ; 1 b 2 a c ) .
Lemma 2.2 Let a , c , λ > 0 , b 0 , 1 2 λ < 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 ) } < a c . 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 , y 0 ), 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 , y 0 ). The lemma is proved. □

Lemma 2.3 Let p and q be conjugate parameters with p > 1 , and let a , c , λ > 0 , b 0 , 1 2 λ < β < 1 , 0 < α 1 < α 2 < π and b 2 max { cos 2 α 1 , cos 2 ( π α 2 ) } < a c , 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 , b 0 , 1 2 λ < β < 1 , 0 < α 1 < α 2 < π and b 2 max { cos 2 α 1 , cos 2 ( π α 2 ) } < a c , and f , g 0 , satisfy 0 < | x | p ( β + 2 λ 2 ) 1 f p ( x ) d x < and 0 < | y | q β 1 g q ( y ) d y < . 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 = K a.e. in  ( , ) × ( , ) .

We suppose A 0 (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 ) d x < . 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 ) d y . 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 ] d x .
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 .

Declarations

Authors’ Affiliations

(1)
Department of Mathematical Analysis, National University of Mongolia
(2)
Institute of Mathematics, National University of Mongolia

References

  1. Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1934.Google Scholar
  2. Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston; 1991.View ArticleGoogle Scholar
  3. Yang B: On Hardy-Hilbert’s integral inequality. J. Math. Anal. Appl. 2001, 261: 295–306. 10.1006/jmaa.2001.7525MathSciNetView ArticleGoogle Scholar
  4. Zeng Z, Xie Z: On a new Hilbert-type integral inequality with the integral in whole plane. J. Inequal. Appl. 2010., 2010: Article ID 256796Google Scholar
  5. Xin D, Yang B: A Hilbert-type integral inequality in the whole plane with the homogeneous kernel of degree −2. J. Inequal. Appl. 2011., 2011: Article ID 401428Google Scholar
  6. Abramowitz M, Stegun IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 9th edition. Dover, New York; 1972:807–808.Google Scholar
  7. Azar LE: Some extension of Hilbert’s integral inequality. J. Math. Inequal. 2011, 5(1):131–140.MathSciNetView ArticleGoogle Scholar
  8. Kuang J: Applied Inequalities. Shangdong Science and Technology Press, Jinan; 2004.Google Scholar

Copyright

© Adiyasuren and Batbold; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.