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.

1. Introduction

If , ,  , and , then we have (cf. [1]) where the constant factor 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: where the constant factor    is the best possible. If , , and   , Yang [14] gave another new Hilbert-type integral inequality in the whole plane in 2008 as follows: where the constant factor 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 , we have , where is the function (cf. [21]). For , , setting  , we find the following expression:

Lemma 1. If , , , and , define the weight functions and as follows: Then, for , we have

Proof. (i) For , setting , we find, for , By Lebesgue term-by-term integration theorem (cf. [22]), in view of (8) and (5), we find
(ii) For , setting , we still can obtain . Setting , we find Since, for ,  , there exists a positive number , such that ;   then, by (9), it follows that and then . Hence we have (7).

Remark 2. We have the following formula of the hypergeometric function (cf. [21]). If , , then In particular, for , , it follows that
In (9), for , in view of (14), we have

Lemma 3. If ,  , ,  , , , is indicated by (7), and is a nonnegative measurable function in , then one has

Proof. By Hölder’s inequality (cf. [23]), we have Then, by (7) and Fubini theorem (cf. [22]), it follows that Hence, in view of (7), inequality (16) follows.

3. Main Results and Applications

Theorem 4. If ,  ,  ,  ,  , and ,  ,  , satisfying and , then one has where the constant factors and are the best possible and is defined by (7). Inequalities (19) and (20) are equivalent.

In particular, for , we have the following equivalent inequalities:

Proof. If (17) takes the form of equality for a , then there exist constants and , such that they are not all zero, and We suppose that (otherwise ). Then it follows that which contradicts the fact that . Hence (17) takes the form of strict inequality and so does (16). Then we have (20). By Hölder’s inequality (cf. [23]), we find By (20), we have (19). On the other hand, suppose that (19) is valid. We set and find . By (16), we have . If , then (20) is obviously value; if , then, by (19), we obtain Hence we have (20), which is equivalent to (19). We indicate two sets and . For , we define two functions , as follows: Then we obtain Since, for , we find and is an even function, then it follows that Setting in the above integral, by Fubini theorem (cf. [22]), we find If the constant factor in (19) is not the best possible, then there exists a positive number with , such that (19) is valid when replacing by . Then we have , and By (8) and Fatou lemma (cf. [22]), we have which contradicts the fact that . Hence the constant factor 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 by , 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 as (25) in Theorem 4, by the reverse of (16), we have . If , then the reverse of (20) is obviously value; if , 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 in the reverse of (19) is not the best possible, then there exists a positive constant , with , such that the reverse of (19) is still valid when replacing by . By the reverse of (32), we have For , by Levi theorem (cf. [22]), we find There exists a constant , such that , and then . For   , since , , and then, by Lebesgue control convergence theorem (cf. [22]), for , we have By (34), (35), and (37), for , we have , which contradicts the fact that . Hence, the constant factor 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 in (19) and (20), replacing by , we obtain the following equivalent inequalities with a homogeneous kernel and the best possible constant factors:
(ii) For    in (19) and (20), we obtain the following equivalent inequalities: where is indicated by (15).
(iii) For ,     in (40), we find and then (3) follows. Hence, (40) and (19) are extensions of (3).

Conflict of Interests

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

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).