Journal of Function Spaces
Volume 2018 (2018), Article ID 2691816, 8 pages
https://doi.org/10.1155/2018/2691816

Research Article

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

Dongmei Xin, Bicheng Yang, and Aizhen Wang

Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, China

Correspondence should be addressed to Bicheng Yang; bcyang818@163.com

Received 22 March 2018; Accepted 16 August 2018; Published 2 September 2018

Academic Editor: Ismat Beg

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.

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 inequality

holds true, then we have

Proof. (i) If , then for , we set two functions

and obtain

By (12) and (14), we find

For 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 obtain

For , , 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 inequality

holds 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 have

Then 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 have

For , 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 (), setting

we 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.

References

G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, Cambridge, USA, 2nd edition, 1988.

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

B. Yang, Hilbert-Type Integral Inequalities, vol. 1, Bentham Science Publishers, The United Arab Emirates, 2009.

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.

J. Xu, “Hardy-Hilbert's inequalities with two parameters,” Advances in Mathematics, vol. 36, no. 2, pp. 189–202, 2007.

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.

L. Debnath and B. Yang, “Recent developments of Hilbert-type discrete and integral inequalities with applications,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 871845, 29 pages, 2012.

Y. Hong, “On the structure character of Hilberts type integral inequality with homogeneous kernal and applications,” Journal of Jilin University (Science Edition), vol. 55, no. 2, pp. 189–194, 2017.

M. T. Rassias and B. Yang, “On half-discrete Hilbert's inequality,” Applied Mathematics and Computation, vol. 220, pp. 75–93, 2013.

M. T. Rassias and B. Yang, “A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function,” Applied Mathematics and Computation, vol. 225, pp. 263–277, 2013.

Q. Huang, “A new extension of a Hardy-Hilbert-type inequality,” Journal of Inequalities and Applications, vol. 2015, article 397, 12 pages, 2015.

B. He and Q. Wang, “A multiple Hilbert-type discrete inequality with a new kernel and best possible constant factor,” Journal of Mathematical Analysis and Applications, vol. 431, no. 2, pp. 889–902, 2015.

M. Krnic' and J. Pecaric', “General Hilberts and Hardys inequalities,” Mathematical Inequalities & Applications, vol. 8, no. 1, pp. 29–51, 2005.

I. Perić and P. Vuković, “Multiple Hilbert's type inequalities with a homogeneous kernel,” Banach Journal of Mathematical Analysis, vol. 5, no. 2, pp. 33–43, 2011.

V. Adiyasuren, Ts. Batbold, and M. Krnić, “Multiple Hilbert-type inequalities involving some differential operators,” Banach Journal of Mathematical Analysis, vol. 10, no. 2, pp. 320–337, 2016.

Q. Chen and B. Yang, “A survey on the study of Hilbert-type inequalities,” Journal of Inequalities and Applications, 2015:302, 29 pages, 2015.

B. Yang, “A new Hilbert's type integral inequality,” Soochow Journal of Mathematics, vol. 33, no. 4, pp. 849–859, 2007.

Z. X. Wang and D. R. Guo, Introduction to Special Functions, Science Press, Beijing, China, 1979.

B. He and B. C. Yang, “A Hilbert-type integral inequality with a homogeneous kernel of 0-degree and a hypergeometric function,” Mathematics in Practice and Theory, vol. 40, no. 18, pp. 203–211, 2010.

Z. Zeng and Z. T. Xie, “On a new hilbert-type intergral inequality with the intergral in whole plane,” Journal of Inequalities and Applications, vol. 2010, Article ID 256796, 8 pages, 2010.

Z. Xie, Z. Zeng, and Y. Sun, “A new Hilbert-type inequality with the homogeneous kernel of degree-2,” Advances and Applications in Mathematical Sciences, vol. 12, no. 7, pp. 391–401, 2013.

Z. Zhen, K. R. R. Gandhi, and 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.

Z. Gu and B. Yang, “A Hilbert-type integral inequality in the whole plane with a non-homogeneous kernel and a few parameters,” Journal of Inequalities and Applications, 2015:314, 9 pages, 2015.

J. C. Kuang, Real and functional analysis (Continuation), vol. second volume, Higher Education Press, Beijing, China, 2015.

J. C. Kuang, Applied Inequalities, Shangdong Science and Technology Press, Jinan, China, 2004.