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 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).