Abstract and Applied Analysis
Volume 2019 (2019), Article ID 8317029, 8 pages
https://doi.org/10.1155/2019/8317029

Research Article

On a Parametric Mulholland-Type Inequality and Applications

Bicheng Yang,¹ Meifa Huang,² and Yanru Zhong³

¹Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, China
²School of Mechanical Science and Electrical Engineering, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China
³School of Computer Science and Information Security, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China

Correspondence should be addressed to Yanru Zhong; 18577399236@163.com

Received 29 March 2019; Accepted 9 May 2019; Published 20 May 2019

Academic Editor: Wing-Sum Cheung

Copyright © 2019 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.

In this paper, by the use of the weight functions, and the idea of introducing parameters, a discrete Mulholland-type inequality with the general homogeneous kernel and the equivalent form are given. The equivalent statements of the best possible constant factor related to a few parameters are provided. As applications, the operator expressions and a few particular examples are considered.

1. Introduction

Assuming that , we have the following discrete Hilbert’s inequality with the best possible constant factor (cf. [1], Theorem 315):

We still have the following Mulholland’s inequality with the same best possible constant (cf. [1], Theorem 343):

If and , then we have the following Hilbert’s integral inequality:

with the best possible constant factor (cf. [1], Theorem 316).

Inequalities (1), (2), and (3) and their extensions with the conjugate exponents and independent parameters are important in analysis and its applications (cf. [2–13]).

The following half-discrete Hilbert-type inequality was provided (cf. [1], Theorem 351). If is decreasing, , then

Some new extensions of (4) were provided by [14–19].

In 2016, by the use of the technique of real analysis, Hong [20] considered some equivalent statements of the extensions of (1) with the best possible constant factor related to a few parameters. The other similar works about the extensions of (3) were provided by [21–25].

In this paper, according to the way given by [20], by the use of the weight functions and the idea of introducing parameters, a discrete Mulholland-type inequality with the general homogeneous kernel and the equivalent form are given, which is an extension of (2). The equivalent statements of the best possible constant factor related to a few parameters are provided. As applications, the operator expressions and a few particular examples are considered.

2. Some Lemmas

In what follows, we suppose that ,   is a positive homogeneous function of degree, satisfying, for any ,

Also, is decreasing with respect to (or , ), such that, for ,

We still assume that , satisfying

Definition 1. Define the following weight functions:

Lemma 2. We have the following inequalities:

Proof. For , it is evident that is a strictly decreasing function with respect to . By the decreasing property, setting , it follows that

Hence, we have (10). For , it is evident that is a strictly decreasing function with respect to . By the decreasing property, setting , we find that

Hence, we have (11).

Lemma 3. We have the following inequality:

Proof. By Hölder’s inequality with weight (cf. [26]), we obtain

Then by (10) and (11), we have (14).

Remark 4. By (14), for , we find

and the following inequality:

In particular, for , we have

For , , , (18) reduces to (2). Hence, (17) is an extension of (18) and (2).

Lemma 5. The constant factor in (17) is the best possible.

Proof. For any , we set

If there exists a constant , such that (17) is valid when replacing by , then, in particular, we have

We obtain

By the decreasing property and Fubini theorem (cf. [27]), we find

Then we have

For , by Fatou lemma (cf. [27]), we find

Hence, is the best possible constant factor of (17).

Remark 6. Setting , , we find

and by Hölder’s inequality (cf. [26]), we have

We can rewrite (14) as follows:

Lemma 7. If the constant factor in (14) is the best possible, then .

Proof. If the constant factor in (14) is the best possible, then, by (27) and (17), the unique best possible constant factor must be , namely,

We observe that (26) keeps the form of equality if and only if there exist constantsand, such that they are not all zero and (cf. [26])

Assuming that (otherwise, ), it follows that in , and then , namely, .

3. Main Results

Theorem 8. Inequality (14) is equivalent to

If the constant factor in (14) is the best possible, then so is the constant factor in (30).

Proof. Suppose that (30) is valid. By Hölder’s inequality (cf. [26]), we find

Then by (30), we obtain (14).
On the other hand, assuming that (14) is valid, we set

If , then (30) is naturally valid; if , then it is impossible to make (30) valid, namely, . Suppose that . By (14), it follows that

namely, (30) follows, which is equivalent to (14).
If the constant factor in (14) is the best possible, then so is constant factor in (30). Otherwise, by (31), we would reach a contradiction that the constant factor in (14) is not the best possible.

Theorem 9. The statements (i), (ii), (iii), and (iv) are equivalent as follows:

(i) is independent of (ii) is expressible as a single integral(iii) is the best possible constant factor of (14)(iv)

If the statement (iv) follows, namely, , then we have (17) and the following equivalent inequality with the best possible constant factor :

Proof. (i)=>(ii). Since is independent of , we find

namely, is expressible as a single integral

(ii)=>(iv). In (26), if is expressible as a single integral , then (26) keeps the form of equality, which follows that .
(iv)=>(i). If , then , which is independent of . Hence, we have (i)⇔(ii)⇔(iv).
(iii)=>(iv). By Lemma 7, we have .
(iv)=>(iii). By Lemma 5, for , is the best possible constant factor of (14). Therefore, we have (iii) ⇔(iv).
Hence, the statements (i), (ii), (iii), and (iv) are equivalent.

Remark 10. (i) For in (17) and (34), we have the following equivalent inequalities with the best possible constant factor :

(ii) For, , in (17) and (34), we have the following equivalent inequalities with the best possible constant factor :

(iii) For , both (37) and (39) reduce to

and both (38) and (40) reduce to the equivalent form of (41) as follows:

4. Operator Expressions and Some Particular Cases

We set functions

where

Define the following real normed spaces:

Assuming that , setting

we can rewrite (30) as follows:

namely, .

Definition 11. Define a Mulholland-type operator as follows: for any , there exists a unique representation . Define the formal inner product of and and the norm of as follows:

By Theorems 8 and 9, we have the following.

Theorem 12. If , then we have the following equivalent inequalities:

Moreover, if and only if the constant factor in (49) and (50) is the best possible, namely,

Example 13. We set . Then we find . For , is a positive homogeneous function of degree , such that is decreasing with respect to, and for ,

In view of Theorem 12, it follows that if and only if

Example 14. We set . Then we find . For , is a positive homogeneous function of degree , such that is decreasing with respect to (cf. [2], Example 2.2.1), and for ,

In view of Theorem 12, it follows that if and only if

Example 15. We set . Then we find

For , is a positive homogeneous function of degree , such that is decreasing with respect to , and for , by Example 1 of [28], it follows that

In view of Theorem 12, it follows that if and only if

In particular, for , we have and

If , then we have , , and

5. Conclusions

In this paper, by the use of the weight functions and the idea of introducing parameters, a discrete Mulholland-type inequality with the general homogeneous kernel and the equivalent form are given in Lemma 3 and Theorem 8. The equivalent statements of the best possible constant factor related to a few parameters are considered in Theorem 9. As applications, the operator expressions and some particular examples are given in Theorem 12 and Examples 13–15. The lemmas and theorems provide an extensive account of this type of inequalities.

Data Availability

The study belongs to pure theory research. There are not any sharing data.

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the National Natural Science Foundation (Nos. 61562016 and 51765012) and Science and Technology Planning Project Item of Guangzhou City (No. 201707010229). We are grateful for this help.

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