Parameterized Hilbert-Type Integral Inequalities in the Whole Plane

Abstract

By the use of the way of real analysis, we estimate the weight functions and give some new Hilbert-type integral inequalities in the whole plane with nonhomogeneous kernels and multiparameters. The constant factors related to the hypergeometric function and the beta function are proved to be the best possible. We also consider the equivalent forms, the reverses, and some particular cases in the homogeneous kernels.

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Qiliang Huang 1 and Shanhe Wu 2 and Bicheng Yang 1

Academic Editor:Tohru Ozawa

1, Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, China
2, Department of Mathematics and Computer Science, Longyan University, Longyan, Fujian 364012, China

Received 21 June 2014; Accepted 5 August 2014; 19 August 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

If f(x) , g(y)...5;0 , 0<^∫0∞ ...^f2 (x)dx<∞ , and 0<^∫0∞ ...^g2 (y)dy<∞ , then we have (cf. [1]) [figure omitted; refer to PDF] where the constant factor π is the best possible. Inequality (1) is well known as Hilbert's integral inequality, which is important in analysis and its applications (cf. [1, 2]). In recent years, by using the way of weight functions, a number of extensions of (1) were given by Yang (cf. [3]). Noticing that inequality (1) is with a homogenous kernel of degree -1, a survey of the study of Hilbert-type inequalities with the homogeneous kernels of degree negative numbers and some parameters was given by [4] in 2009. Recently, some inequalities with the homogenous kernels and nonhomogenous kernels have been studied (cf. [5-12]). All of the above integral inequalities are built in the quarter plane of the first quadrant.

In 2007, Yang [13] first gave a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane as follows: [figure omitted; refer to PDF] where the constant factor B(λ/2,λ/2) (λ>0) is the best possible. If 0<λ<1 , p>1 , and (1/p)+(1/q)=1 , Yang [14] gave another new Hilbert-type integral inequality in the whole plane in 2008 as follows: [figure omitted; refer to PDF] where the constant factor [figure omitted; refer to PDF] is the best possible. He et al. [15-20] also provided some Hilbert-type integral inequalities in the whole plane by using some new methods and techniques.

In this paper, by the use of the way of real analysis, we estimate the weight functions and give some new Hilbert-type integral inequalities in the whole plane with nonhomogeneous kernels and multiparameters, which are extensions of (3). The constant factors related to the hypergeometric function and the beta function are proved to be the best possible. We also consider the equivalent forms, the reverses, and some particular inequalities with the homogeneous kernels.

2. Some Lemmas

Assuming that α>0 , we have Γ(α)=^∫0∞ ...^xα-1e-x dx , where Γ(α) is the Γ function (cf. [21]). For β>-1 , η>0 , setting v=-η ln......t , we find the following expression: [figure omitted; refer to PDF]

Lemma 1.

If β>-1 , min......{μ,σ}>-α , μ+σ=λ<1+β , and δ∈{-1,1} , define the weight functions _ωδ (σ,y) and _[varpi]δ (σ,x) as follows: [figure omitted; refer to PDF] Then, for y,x∈(-∞,0)∪(0,+∞) , we have [figure omitted; refer to PDF]

Proof.

(i) For δ=1 , setting u=xy , we find, for y∈(-∞,0)∪(0,+∞) , [figure omitted; refer to PDF] By Lebesgue term-by-term integration theorem (cf. [22]), in view of (8) and (5), we find [figure omitted; refer to PDF]

(ii) For δ=-1 , setting y/x , we still can obtain _ω-1 (σ,y)=_Kβ (σ) . Setting u=^xδ y , we find [figure omitted; refer to PDF] Since, for β>-1 , 0<_θ0 <min...{μ+α,σ+α} , [figure omitted; refer to PDF] there exists a positive number L , such that ((-ln...u)/(1-u)^)βu_θ0 ...4;L(u∈(0,1]) ; then, by (9), it follows that [figure omitted; refer to PDF] and then _Kβ (σ)∈_R+ . Hence we have (7).

Remark 2.

We have the following formula of the hypergeometric function F (cf. [21]). If Re(γ)>Re(θ)>0 , |arg(1-z)|<π , then [figure omitted; refer to PDF] In particular, for z=-1 , γ=θ+1 (θ>0) , it follows that [figure omitted; refer to PDF]

In (9), for β=0 (λ<1) , in view of (14), we have [figure omitted; refer to PDF]

Lemma 3.

If p>1 , (1/p)+(1/q)=1 , β>-1 , min...{μ,σ}>-α , μ+σ=λ<1+β , δ∈{-1,1} , _Kβ (σ) is indicated by (7), and f(x) is a nonnegative measurable function in (-∞,∞) , then one has [figure omitted; refer to PDF]

Proof.

By Hölder's inequality (cf. [23]), we have [figure omitted; refer to PDF] Then, by (7) and Fubini theorem (cf. [22]), it follows that [figure omitted; refer to PDF] Hence, in view of (7), inequality (16) follows.

3. Main Results and Applications

Theorem 4.

If p>1 , (1/p)+(1/q)=1 , β>-1 , min...{μ,σ}>-α , μ+σ=λ<1+β , and δ∈{-1,1} , f(x) , g(y)...5;0 , satisfying 0<^∫-∞∞ ...^{|x|p(1-δσ)-1fp} (x)dx<∞ and 0<^∫-∞∞ ...|y^|q(1-σ)-1gq (y)dy<∞ , then one has [figure omitted; refer to PDF] where the constant factors _Kβ (σ) and ^Kβp (σ) are the best possible and _Kβ (σ) is defined by (7). Inequalities (19) and (20) are equivalent.

In particular, for δ=1 , we have the following equivalent inequalities: [figure omitted; refer to PDF]

Proof.

If (17) takes the form of equality for a y∈(-∞,0)∪(0,∞) , then there exist constants A and B , such that they are not all zero, and [figure omitted; refer to PDF] We suppose that A...0;0 (otherwise B=A=0 ). Then it follows that [figure omitted; refer to PDF] which contradicts the fact that 0<^∫-∞∞ ...^{|x|p(1-δσ)-1fp} (x)dx<∞ . Hence (17) takes the form of strict inequality and so does (16). Then we have (20). By Hölder's inequality (cf. [23]), we find [figure omitted; refer to PDF] By (20), we have (19). On the other hand, suppose that (19) is valid. We set [figure omitted; refer to PDF] and find J=^∫-∞∞ ...|y^|q(1-σ)-1gq (y)dy . By (16), we have J<∞ . If J=0 , then (20) is obviously value; if 0<J<∞ , then, by (19), we obtain [figure omitted; refer to PDF] Hence we have (20), which is equivalent to (19). We indicate two sets _Eδ [: =]{x∈R;|x^|δ ...5;1} and ^Eδ+ [: =]_Eδ ∩_R+ ={x∈_R+ ;^xδ ...5;1} . For [varepsilon]>0 , we define two functions f~(x) , g~(y) as follows: [figure omitted; refer to PDF] Then we obtain [figure omitted; refer to PDF] Since, for Y=-y , we find [figure omitted; refer to PDF] and h(x) is an even function, then it follows that [figure omitted; refer to PDF] Setting v=^xδ in the above integral, by Fubini theorem (cf. [22]), we find [figure omitted; refer to PDF] If the constant factor _Kβ (σ) in (19) is not the best possible, then there exists a positive number k with _Kβ (σ)<k , such that (19) is valid when replacing _Kβ (σ) by k . Then we have I~<kL~ , and [figure omitted; refer to PDF] By (8) and Fatou lemma (cf. [22]), we have [figure omitted; refer to PDF] which contradicts the fact that k<_Kβ (σ) . Hence the constant factor _Kβ (σ) in (19) is the best possible. If the constant factor in (20) is not the best possible, then, by (24), we may get a contradiction that the constant factor in (19) is not the best possible.

Theorem 5.

As the assumptions of Theorem 4, replacing p>1 by 0<p<1 , one has the equivalent reverses of (19) and (20) with the same best constant factors.

Proof.

By the reverse Hölder's inequality (cf. [23]), we have the reverses of (16) and (24). It is easy to obtain the reverse of (20). In view of the reverses of (20) and (24), we obtain the reverse of (19). On the other hand, suppose that the reverse of (19) is valid. Setting the same g(y) as (25) in Theorem 4, by the reverse of (16), we have J>0 . If J=∞ , then the reverse of (20) is obviously value; if J<∞ , then, by the reverse of (19), we obtain the reverses of (26). Hence we have the reverse of (20), which is equivalent to the reverse of (19). If the constant factor _Kβ (σ) in the reverse of (19) is not the best possible, then there exists a positive constant k , with k>_Kβ (σ) , such that the reverse of (19) is still valid when replacing _Kβ (σ) by k . By the reverse of (32), we have [figure omitted; refer to PDF] For [varepsilon][arrow right]⁰⁺ , by Levi theorem (cf. [22]), we find [figure omitted; refer to PDF] There exists a constant _δ0 >0 , such that σ-(1/2)_δ0 >-α , and then 0<_Kβ (σ-(_δ0 /2))<∞ . For 0<[varepsilon]<_δ0 |q|/4 (q<0) , since ^{uσ+α+(2[varepsilon]/q)-1} ...4;^{uσ+α-(_δ0 /2)-1} , u∈(0,1] , and [figure omitted; refer to PDF] then, by Lebesgue control convergence theorem (cf. [22]), for [varepsilon][arrow right]⁰⁺ , we have [figure omitted; refer to PDF] By (34), (35), and (37), for [varepsilon][arrow right]⁰⁺ , we have _Kβ (σ)...5;k , which contradicts the fact that k>_Kβ (σ) . Hence, the constant factor _Kβ (σ) in the reverse of (19) is the best possible. If the constant factor in the reverse of (20) is not the best possible, then, by the reverse of (24), we may get a contradiction that the constant factor in the reverse of (19) is not the best possible.

Remark 6.

(i) For δ=-1 in (19) and (20), replacing |x^|λ f(x) by f(x) , we obtain the following equivalent inequalities with a homogeneous kernel and the best possible constant factors: [figure omitted; refer to PDF]

(ii) For β=0 (λ<1) in (19) and (20), we obtain the following equivalent inequalities: [figure omitted; refer to PDF] where _K0 (σ) is indicated by (15).

(iii) For α=0 , σ=μ=λ/2 (0<λ<1) in (40), we find [figure omitted; refer to PDF] and then (3) follows. Hence, (40) and (19) are extensions of (3).

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 61370186), 2013 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (no. 2013KJCX0140), and the Foundation of Scientific Research Project of Fujian Province Education Department of China (no. JK2012049).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] G. H. Hardy, J. E. Littlewood, G. Po'lya Inequalities , Cambridge University Press, Cambridge, Mass, USA, 1934.

[2] D. S. Mitrinovic, J. E. Pecaric, A. M. Fink Inequalities Involving Functions and Their Integrals and Derivatives , vol. 53, of Boston, Mass, USA, Kluwer Academic Publishers, 1991.

[3] B. Yang The Norm of Operator and Hilbert-Type Inequalities , Science Press, Beijing, China, 2009.

[4] B. C. Yang, "A survey of the study of Hilbert-type inequalities with parameters," Advances in Mathematics , vol. 38, no. 3, pp. 257-268, 2009.

[5] B. Yang, "On the norm of an integral operator and applications," Journal of Mathematical Analysis and Applications , vol. 321, no. 1, pp. 182-192, 2006.

[6] I. Brnetic, J. Pecaric, "Generalizations of Hilbert's integral inequality," Mathematical Inequalities and Applications , vol. 7, no. 2, pp. 199-205, 2004.

[7] M. Krnic, J. Pecaric, "General Hilbert's and Hardy's inequalities," Mathematical Inequalities & Applications , vol. 8, no. 1, pp. 29-51, 2005.

[8] J. Xu, "Hardy-Hilbert's inequalities with two parameters," Advances in Mathematics , vol. 36, no. 2, pp. 189-202, 2007.

[9] B. Yang, "On the norm of a Hilbert's type linear operator and applications," Journal of Mathematical Analysis and Applications , vol. 325, no. 1, pp. 529-541, 2007.

[10] D. M. Xin, "A Hilbert-type integral inequality with a homogeneous kernel of zero degree," Mathematical Theory and Applications , vol. 30, no. 2, pp. 70-74, 2010.

[11] Q. Liu, W. Sun, "A Hilbert-type integral inequality with multiparameters and a nonhomogeneous kernel," Abstract and Applied Analysis , vol. 2014, 2014.

[12] L. Debnath, B. Yang, "Recent developments of Hilbert-type discrete and integral inequalities with applications," International Journal of Mathematics and Mathematical Sciences , vol. 2012, 2012.

[13] B. Yang, "A new Hilbert's type integral inequality," Soochow Journal of Mathematics , vol. 33, no. 4, pp. 849-859, 2007.

[14] B. C. Yang, "A new Hilbert-type integral inequality with some parameters," Journal of Jilin University (Science Edition) , vol. 46, no. 6, pp. 1085-1090, 2008.

[15] B. He, B. C. Yang, "A Hilbert-type integral inequality with a homogeneous kernel of 0-degree and a hypergeometric function," Mathematics in Practice and Theory. Shuxue de Shijian yu Renshi , vol. 40, no. 18, pp. 203-211, 2010.

[16] Z. Zeng, Z. Xie, "On a new Hilbert-type integral inequality with the homogeneous kernel of degree 0 and the integral in whole plane," Journal of Inequalities and Applications , vol. 2010, 2010.

[17] A. Wang, B. Yang, "A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel," Journal of Inequalities and Applications , vol. 2011, article 123, 2011.

[18] B. He, B. Yang, "On an inequality concerning a non-homogeneous kernel and the hypergeometric function," Tamsui Oxford Journal of Information and Mathematical Sciences , vol. 27, no. 1, pp. 75-88, 2011.

[19] Z. Xie, Z. Zeng, Y. Sun, "A new Hilbert-type inequality with the homogeneous kernel of degree," Advances and Applications in Mathematical Sciences , vol. 12, no. 7, pp. 391-401, 2013.

[20] Z. Zhen, K. R. R. Gandhi, Z. Xie, "A new Hilbert-type inequality with the homogeneous kernel of degree-2 and with the integral," Bulletin of Mathematical Sciences & Applications , vol. 3, no. 1, pp. 11-20, 2014.

[21] Z. Wang, D. Guo Introduction to Special Functions , Science Press, Beijing, China, 1979.

[22] J. Kuang Introudction to Real Analysis , Hunan Educiton Press, Changsha, China, 1996.

[23] J. Kuang Applied Inequalities , Shangdong Science and Technology Press, Jinan, China, 2004.