On a Parametric Mulholland-Type Inequality and Applications

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, , thenSome 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 thatHence, we have (10). For , it is evident that is a strictly decreasing function with respect to . By the decreasing property, setting , we find thatHence, we have (11).

Lemma 3. We have the following inequality:

Proof. By Hölder’s inequality with weight (cf. [26]), we obtainThen by (10) and (11), we have (14).

Remark 4. By (14), for , we findand the following inequality:In particular, for , we haveFor , , , (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 setIf there exists a constant , such that (17) is valid when replacing by , then, in particular, we haveWe obtainBy the decreasing property and Fubini theorem (cf. [27]), we findThen we haveFor , by Fatou lemma (cf. [27]), we findHence, is the best possible constant factor of (17).

Remark 6. Setting , , we findand 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 setIf , then (30) is naturally valid; if , then it is impossible to make (30) valid, namely, . Suppose that . By (14), it follows thatnamely, (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 findnamely, 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 toand both (38) and (40) reduce to the equivalent form of (41) as follows:

4. Operator Expressions and Some Particular Cases

We set functionswhere Define the following real normed spaces:

Assuming that , settingwe 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 thatIn view of Theorem 12, it follows that if and only if In particular, for , we have andIf , 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.