Equivalent Property of a Hilbert-Type Integral Inequality Related to the Beta Function in the Whole Plane

Xin, Dongmei; Yang, Bicheng; Wang, Aizhen

doi:https://doi.org/10.1155/2018/2691816

Journal of Function Spaces

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Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 2691816 | https://doi.org/10.1155/2018/2691816

Equivalent Property of a Hilbert-Type Integral Inequality Related to the Beta Function in the Whole Plane

Dongmei Xin,¹Bicheng Yang,¹and Aizhen Wang¹

Academic Editor: Ismat Beg

Received22 Mar 2018

Accepted16 Aug 2018

Published02 Sept 2018

Abstract

By means of the technique of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The constant factor related to the beta function is proved to be the best possible. As applications, the case of the homogeneous kernel, the operator expressions, and a few corollaries are considered.

1. Introduction

Suppose that , , and . We have the following well-known Hardy-Hilbert’s integral inequality (see [1]):where the constant factor is the best possible. For , (1) reduces to the well-known Hilbert’s integral inequality. By using the weight functions, some extensions of (1) were given by [2, 3]. A few Hilbert-type inequalities with the homogenous and nonhomogenous kernels were provided by [4–7]. In 2017, Hong [8] also gave two equivalent statements between Hilbert-type inequalities with the general homogenous kernel and parameters. Some other kinds of Hilbert-type inequalities were obtained by [9–16].

In 2007, Yang [17] gave a Hilbert-type integral inequality in the whole plane as follows:with the best possible constant factor (, is the beta function) (see [18]). He et al. [19–23] proved a few Hilbert-type integral inequalities in the whole plane with the best possible constant factors.

In this paper, by means of the technique of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane similar to (2) are obtained. The constant factor related to the beta function is proved to be the best possible. As applications, the case of the homogeneous kernel, the operator expressions, and a few corollaries are considered.

2. An Example and Two Lemmas

Example 1. For , we set , and then for ,For , in view of , we find where is the beta function (cf. [18]).

In particular, (i) for , we have , and

(ii) for , we have , and

(iii) for , we have , , and In the case of (iii), for , we have , and

In the following, we assume that , , and

For , we define two sets , and the following two expressions:

Setting in (10), in view of Fubini theorem (cf. [24]), it follows that

In the same way, we find that

Lemma 2. If there exists a constant , such that for any nonnegative measurable functions and in , the following inequalityholds true, then we have

Proof. (i) If , then for , we set two functions and obtain By (12) and (14), we findFor any , it follows that In view of , by (17), we find that , which is a contradiction.
(ii) If , then for , we set functions and find By (13) and (14), we obtainFor , , it follows that By (20), in view of , we have , which is a contradiction.
Hence, we conclude that
The lemma is proved.

For , we have the following.

Lemma 3. If there exists a constant , such that for any nonnegative measurable functions and in , the following inequalityholds true, then we have

Proof. By (12), for , we obtain We use inequality (for ) as follows:By Fatou lemma (cf. [24]) and (23), it follows that The lemma is proved.

3. Main Results and Some Corollaries

Theorem 4. If is a constant, then the following statements (i), (ii), and (iii) are equivalent:
(i) For any nonnegative measurable function in , we have the following inequality:(ii) For any nonnegative measurable functions and in , we have the following inequality:(iii) , and

Proof. . By Hölder’s inequality (see [25]), we haveThen by (25), we have (26).
. By Lemma 2, we have Then by Lemma 3, we have .
. Setting , we obtain the following weight functions: for ,By Hölder’s inequality with weight and (28), we haveFor , by Fubini theorem (see [24]) and (29), we have For , we have (25).
Therefore, the statements (i), (ii), and (iii) are equivalent.
The theorem is proved.

Theorem 5. The following statements (i) and (ii) are valid and equivalent:
(i) For any , satisfying , we have the following inequality: (ii) For any , satisfying and , satisfying , we have the following inequality:

Moreover, the constant factor in (32) and (33) is the best possible.

In particular, for we have the following equivalent inequalities with the best possible constant factor :

Proof. We first prove that (32) is valid. If (30) takes the form of equality for a , then (see [25]), 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, (30) takes the form of strict inequality. For by the proof of Theorem 4, we obtain (32).
. By (27) (for ) and (32), we have (33).
. We set the following function: If , then it is impossible since (32) is valid; if , then (32) is trivially valid. In the following, we suppose that By (33), we have namely, (32) follows, which is equivalent to (33).
Hence, Statements (i) and (ii) are valid and equivalent.
If there exists a constant , such that (33) is valid when replacing by , then by Lemma 3, we have Hence, the constant factor in (33) is the best possible.
The constant factor in (32) is still the best possible. Otherwise, by (27) (for ), we would reach a contradiction that the constant factor in (33) is not the best possible.
The theorem is proved.

For , and in Theorems 4 and 5, then replacing () by (), settingwe have the following corollaries.

Corollary 6. If is a constant, then the following statements (i), (ii), and (iii) are equivalent:
(i) For any nonnegative measurable function in , we have the following inequality:(ii) For any nonnegative measurable functions and in , we have the following inequality:(iii) , and

Corollary 7. The following statements (i) and (ii) are valid and equivalent:
(i) For any , satisfying , we have the following inequality:(ii) For any , satisfying , and , satisfying , we have the following inequality:Moreover, the constant factor in (44) and (45) is the best possible.
In particular, for , , we have the following equivalent inequalities with the best possible constant factor :

In (35) and (36), setting , then replacing back by , and introducing the hyperbolic sine function as , we have

Corollary 8. If , , then the following statements (i) and (ii) are valid and equivalent:
(i) For any , satisfying , we have the following inequality:(ii) For any , satisfying and , satisfying , we have the following inequality:Moreover, the constant factor in (48) and (49) is the best possible.

4. Operator Expressions

We set the following functions: , wherefrom, , and define the following real normed linear spaces:wherefrom,

(a) In view of Theorem 5, for , setting by (34), we have

Definition 9. Define a Hilbert-type integral operator with the nonhomogeneous kernel as follows: for any , there exists a unique representation , satisfying for any
In view of (53), it follows that and then the operator is bounded satisfying If we define the formal inner product of and as follows: then we can rewrite Theorem 5 as follows.

Theorem 10. The following statements (i) and (ii) are valid and equivalent:
(i) For any , satisfying , we have the following inequality:(ii) For any , satisfying , and , we have the following inequality:Moreover, the constant factor in (57) and (58) is the best possible, namely,

(b) In view of Corollary 7, for , setting by (44), we have

Definition 11. Define a Hilbert-type integral operator with the homogeneous kernel as follows: for any , there exists a unique representation , satisfying for any
In view of (61), it follows that and then the operator is bounded satisfying

If we define the formal inner product of and as follows: then we can rewrite Corollary 7 as follows.

Corollary 12. The following statements (i) and (ii) are valid and equivalent:
(i) For any , satisfying , we have the following inequality:(ii) For any , satisfying , and , we have the following inequality:Moreover, the constant factor in (65) and (66) is the best possible, namely,

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation (nos. 61370186 and 61640222) and Science and Technology Planning Project Item of Guangzhou City (no. 201707010229). We are grateful for this help.

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Copyright © 2018 Dongmei Xin et al. 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.

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