Abstract

By the use of weight functions and technique of real analysis, a new half-discrete Hilbert-type inequality in the whole plane with multiparameters and the best possible constant factor is given. Furthermore, the equivalent forms, two kinds of particular inequalities, and the operator expressions with the norm are considered.

1. Introduction

If , then we have the following discrete Hardy-Hilbert inequality (cf. [1]):where the constant factor is the best possible one. Assuming that , satisfying and , we have the following Hardy-Hilbert integral inequality (cf. [2]):with the best possible constant factor . Recently, half-discrete Hardy-Hilbert’s inequality with the same best possible constant factor was given as follows [3]:Inequalities (1), (2), and (3) are important in analysis and its applications (cf. [2, 4, 5]).

Noticing that inequalities (1) and (2) are with homogenous kernels of degree −1, in 2009, a survey of the study of Hilbert-type inequalities with the homogeneous kernels of degree negative numbers and some parameters is given in [6]. Recently, some inequalities with the homogenous kernels of degree 0 and nonhomogenous kernels have been studied in [7, 8]. The other kinds of Hilbert-type inequalities are provided in [9–19]. All of the above inequalities are built in the quarter plane of the first quadrant.

In 2007, Yang [20] gave a new Hilbert-type integral inequality in the whole plane as follows:where the constant factor is the best possible one ( is the beta function). And Zeng et al. [21, 22] also published some new Hilbert-type integral inequalities in the whole plane.

In this paper, by the use of weight functions and technique of real analysis, a new half-discrete Hilbert-type inequality in the whole plane with the best possible constant factor is built as follows: for , ,Furthermore, an extension of (5) with multiparameters is given. The equivalent forms, two kinds of particular inequalities, and the operator expressions with the norm are considered.

2. Some Lemmas

In the following, we agree that : wherefrom

Lemma 1 (cf. [23]). Suppose that is decreasing in and strictly decreasing in , satisfying One has

Lemma 2. One defines two weight functions and as follows:where . Then, for , one hasfor , one haswhere

Proof. (i) We have Setting in the above first (second) integral, by simplifications, we find Hence, we have (11).
(ii) We haveSince, for , both and are strictly decreasing in , satisfying by (16) and (8), we obtain Setting in the above first (second) integral, by simplifications, we find By (16) and (8), we still have Setting in the above first (second) integral, by simplifications, we obtain where is indicated by (13). We find and then (12) and (13) follow.

Lemma 3. For , setting , one has

Proof. Setting it follows that We find Setting in the above first (second) integral, we obtain Hence we have (23).

Lemma 4. For , setting we have

Proof. We haveBy (29) and (8), we find Hence, we have (28).

3. Main Results

Theorem 5. Suppose that , , , andIf , , satisfying then we have the following equivalent inequalities:In particular, for , we have the following equivalent inequalities:

Proof. By Hölder’s inequality (cf. [24]) and (9), we find Then by (11) and Lebesgue term-by-term integration theorem (cf. [25]), in view of (10), we findThen, by (12), we have (34).
By Hölder’s inequality (cf. [24]), we haveThen, by (34), we have (33). On the other hand, assuming that (33) is valid, we set Then we find In view of (40), it follows that If , and then (34) is trivially valid; if , then, by (33), we have namely, (34) holds, which is equivalent to (33).
In the same way of obtaining (40), we haveWe have proved that (33) is valid. Setting it follows that and, in view of (45), If , then (35) is trivially valid; if , then, by (33), we have namely, (35) follows.
On the other hand, assuming that (35) is valid, by Hölder’s inequality (cf. [24]) and in the same way of obtaining (41), we haveThen, by (35), we have (33), which is equivalent to (35).
Therefore, inequalities (33), (34), and (35) are equivalent.

Theorem 6. As regards the assumptions of Theorem 5, the constant factor in (33), (34), and (35) is the best possible one.

Proof. For , we set , and Then, by (23) and (28), we find By (12) and (23), we still have If the constant factor in (33) is not the best possible one, then, there exists a positive number with , such that (33) is still valid when replacing by . Then, in particular, we have ; namely, It follows that , and then which contradicts the fact that . Hence, the constant factor in (33) is the best possible one.
The constant factor in (34) ((35)) is still the best possible one. Otherwise, we would reach a contradiction by (41) ((49)) that the constant factor in (33) is not the best possible one.

4. Operator Expressions

Suppose that and . We set the following functions: wherefrom and Define the following real weight normed linear spaces:

(a) In view of Theorem 5, for , setting by (34), we havenamely,

Definition 7. Define a half-discrete Hilbert-type operator in the whole plane as follows: for any , there exists a unique representation , satisfying, for any , .
In view of (59), it follows that , and then the operator is bounded satisfying Since the constant factor in (59) is the best possible one, we haveIf we define the formal inner product of and as follows: then we can rewrite (33) and (34) as follows:(b) In view of Theorem 5, for , setting then, by (35), we havenamely, .

Definition 8. Define a half-discrete Hilbert-type operator in the whole plane as follows: for any , there exists a unique representation , satisfying, for any , .
In view of (65), it follows that , and then the operator is bounded, satisfying Since the constant factor in (65) is the best possible one, we haveIf we define the formal inner product of and as follows: then we can rewrite (33) and (35) as follows:

Remark 9. (i) For , (36) reduces to (5). If and , then (5) reduces to the following half-discrete Hilbert-type inequality:(ii) For , (33) reduces to the following particular inequality with the homogeneous kernel of degree 0:(iii) For , (33) reduces to the following particular inequality with the nonhomogeneous kernel: The constant factors in (5) and the above inequalities are all the best possible ones.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

Bicheng Yang carried out the mathematical studies, participated in the sequence alignment, and drafted the manuscript. Qunwei Ma and Leping He participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by Jishou University Graduate Student Research Innovation Project in 2016 (no. JGY201644) and Hunan Province Natural Science Foundation (no. 2015JJ4041). The authors are grateful for their help.