Journal of Function Spaces

Volume 2018 (2018), Article ID 9569380, 8 pages

https://doi.org/10.1155/2018/9569380

## On a New Extension of Mulholland’s Inequality in the Whole Plane

^{1}Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 51003, China^{2}Guangxi Colleges and Universities Key Laboratory of Intelligent Processing of Computer Image and Graphics, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China^{3}Department of Computer Science, Guangdong University of Education, Guangzhou, Guangdong 51003, China

Correspondence should be addressed to Yanru Zhong

Received 4 November 2017; Accepted 12 December 2017; Published 1 February 2018

Academic Editor: Pasquale Vetro

Copyright © 2018 Bicheng Yang 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

A new, more accurate extension of Mulholland’s inequality in the whole plane with a best possible constant factor is presented by introducing independent parameters, applying weight coefficients and using Hermite-Hadamard’s inequality. Moreover, the equivalent forms, some particular cases, and the operator expressions are considered.

#### 1. Introduction

Assume that , , , , and ; then Hardy-Hilbert’s inequality (cf. [1]) is as follows:where is the best possible constant factor. Replacing and by* a*_{m} and* b*_{n}, respectively, yields the following Mulholland’s inequality (cf. Theorem in [1]):(1) and (2) are two important inequalities in analysis and its applications (cf. [1, 2]).

In 2007, Yang [3] firstly provided the following Hilbert-type integral inequality in the whole plane:where is the best possible constant factor. Various extensions of inequalities (1)–(3) (cf. [4–15]) have been presented since then. Recently, Yang and Chen [16] presented an extension of (1) in the whole plane as follows:where is the best possible constant factor. In addition to Yang and Chen, Xin et al. [17] have also carried out a similar work.

In this paper, we present a new, more accurate extension of (2) in the whole plane with a best possible constant factor that is similar to that in (4) via introducing independent parameters, applying weight coefficients, and using Hermite-Hadamard’s inequality. Moreover, the equivalent forms, some particular cases, and the operator expressions are considered.

#### 2. Two Lemmas

In this section, we assume that , , , , ,

*Remark 1. *In view of the conditions that , it follows thatFor , let the functionWe define two weight coefficients as follows:where

Lemma 2. *The inequalitiesare valid, where*

*Proof. *For , let andThenyieldsIn virtue of , , we find that, for ,and it follows that are strict decreasing and strict convex in . Then Hermite-Hadamard’s inequality (cf. [18]) and (14) yieldSetting in the above first (second) integral, in view of Remark 1 we obtainby simplifications. Similarly, (14) yieldsaccording to monotonicity, where is indicated by (11). It follows that andHence, (10) and (11) are valid.

Similarly, we have the following.

Lemma 3. *For , , the inequalitiesare valid, where*

Lemma 4. *If , then we have*

*Proof. *According to Hermite-Hadamard’s inequality, we obtainThis leads toTherefore, (23) is valid.

#### 3. Main Results

In this section, we also define , and

Theorem 5. *Suppose that ,Then we obtain the following equivalent inequalities:Particularly, (i) for , , we have**(ii) For , , we have*

*Proof. *According to Hölder’s inequality with weight (cf. [18]) and (9), we find thatThen (21) yieldsCombining (10) and (26), we obtain (29).

Using Hölder’s inequality again, we obtainThen according to (29), we obtain (28).

Further, assume that (28) is valid; letand findAccording to (34), it follows that If , then (30) is trivially valid; if , then we haveThus, (29) is valid, which is equivalent to (28).

Theorem 6. *With regard to the assumptions in Theorem 5, is the best possible constant factor in (28) and (29).*

*Proof. *For , let , , andThen (23) and (21) yieldIf there exists a positive number , such that (28) is still valid when replacing by , then we obtainHence, in view of the above results, it follows thatand thennamely,Hence, is the best possible constant factor in (28).

in (29) is the best possible constant factor. Otherwise, we would obtain a contradiction according to (35) that in (28) is not the best possible constant factor.

#### 4. Operator Expressions

Let , and , whereWe define the real weighted normed function spaces as follows:For , let and , and it follows from (29) that ; namely,

Further, we define a Mulholland-type operator as follows: for , , there exists a unique representation We also define the following formal inner product of and :Hence, we can, respectively, rewrite (28) and (29) as the following operator expressions:It follows that the operator is bounded withSince in (29) is the best possible constant factor, we obtain

*Remark 7. *(i) For in (30), we have the following new inequality:It follows that (30) is a more accurate extension of (51).

(ii) If , then (30) reduces to(iii) If , , , , then (52) reduces toFor , (53) reduces to (2). Hence, (28) is a new extension of (2).

#### 5. Conclusions

In this paper, we proposed a new, more accurate extension of Mulholland’s inequality in the whole plane with the best possible constant factor. The equivalent forms, a few particular cases, and the operator expressions were considered and described as some lemmas and theorems in the extension. The method of the real analysis is very important and is the key to prove the equivalent inequalities with the best possible constant factor. The lemmas and theorems can provide an extensive account of this type of inequalities.

#### 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, 61640222, and 61562016) and Science and Technology Planning Project Item of Guangzhou City (no. 201707010229).

#### References

- G. H. Hardy, J. E. Littlewood, and G. Polya,
*Inequalities*, Cambridge University Press, 1934. View at MathSciNet - D. S. Mitrinović, J. E. Pečarić, and A. M. Fink,
*Inequalities involving functions and their integrals and derivatives*, vol. 53 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. View at Publisher · View at Google Scholar · View at MathSciNet - B. Yang, “A new Hilbert's type integral inequality,”
*Soochow Journal of Mathematics*, vol. 33, no. 4, pp. 849–859, 2007. View at Google Scholar · View at MathSciNet - Y. Hong, “All-sided generalization about Hardy-Hilbert integral inequalities,”
*Acta Mathematica Sinica*, vol. 44, no. 4, pp. 619–626, 2001. View at Google Scholar · View at MathSciNet - L. E. Azar, “On some extensions of Hardy-Hilbert's inequality and applications,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 546829, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - M. Krnić and J. Pecarić, “General Hilbert's and Hardy's inequalities,”
*Mathematical Inequalities & Applications*, vol. 8, no. 1, pp. 29–51, 2005. View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - V. Adiyasuren, T. 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. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Li and B. He, “On inequalities of Hilbert's type,”
*Bulletin of the Australian Mathematical Society*, vol. 76, no. 1, pp. 1–13, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W. Zhong, “A Hilbert-type linear operator with the norm and its applications,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 494257, 2009. View at Publisher · View at Google Scholar · View at Scopus - Q. Huang, “On a Multiple hilbert's inequality with parameters,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 309319, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Krnić and P. Vuković, “On a multidimensional version of the Hilbert type inequality,”
*Analysis Mathematica*, vol. 38, no. 4, pp. 291–303, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Q. Huang, “A new extension of a Hardy-Hilbert-type inequality,”
*Journal of Inequalities and Applications*, vol. 2015, article 397, 12 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Q. Huang and B. Yang, “A more accurate half-discrete hilbert inequality with a nonhomogeneous kernel,”
*journal of function spaces and applications*, vol. 2013, Article ID 628250, 2013. View at Publisher · View at Google Scholar · View at Scopus - B. Yang and Q. Chen, “A new extension of Hardy-Hilbert's inequality in the whole plane,”
*Journal of Function Spaces*, vol. 2016, Article ID 9197476, pp. 1–8, 2016. View at Publisher · View at Google Scholar · View at Scopus - D. Xin, B. Yang, and Q. Chen, “A discrete Hilbert-type inequality in the whole plane,”
*Journal of Inequalities and Applications*, vol. 2016, no. 1, article no. 133, 2016. View at Publisher · View at Google Scholar · View at Scopus - J. Kuang,
*Applied inequalities*, Shangdong Science Technic Press, Jinan, China, 2010.