Journal of Function Spaces
Volume 2018 (2018), Article ID 9569380, 8 pages
https://doi.org/10.1155/2018/9569380

Research Article

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

Bicheng Yang,¹ Yanru Zhong,² and Qiang Chen³

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

Correspondence should be addressed to Yanru Zhong; rosezhong@guet.edu.cn

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 am and bn, 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 that

For , let the function

We define two weight coefficients as follows:

where

Lemma 2. The inequalities

are valid, where

Proof. For , let and

Then

yields

In 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) yield

Setting in the above first (second) integral, in view of Remark 1 we obtain

by simplifications. Similarly, (14) yields

according to monotonicity, where is indicated by (11). It follows that and

Hence, (10) and (11) are valid.

Similarly, we have the following.

Lemma 3. For , , the inequalities

are valid, where

Lemma 4. If , then we have

Proof. According to Hermite-Hadamard’s inequality, we obtain

This leads to

Therefore, (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 that

Then (21) yields

Combining (10) and (26), we obtain (29).
Using Hölder’s inequality again, we obtain

Then according to (29), we obtain (28).
Further, assume that (28) is valid; let

and find

According to (34), it follows that If , then (30) is trivially valid; if , then we have

Thus, (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 , , and

Then (23) and (21) yield

If there exists a positive number , such that (28) is still valid when replacing by , then we obtain

Hence, in view of the above results, it follows that

and then

namely,

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 , where

We 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 with

Since 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 to

For , (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).

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