Abstract
By introducing multi-parameters, applying the weight coefficients and Hermite–Hadamard’s inequality, we give a reverse of the extended Mulholland inequality in the whole plane with the best possible constant factor. The equivalent forms and a few particular cases are also considered.
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1 Introduction
Assuming that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(a_{n},b_{n}\geq0\), \(0<\sum_{n=1}^{\infty}a_{n}^{p}<\infty\), and \(0<\sum_{n=1}^{\infty }b_{n}^{q}<\infty\), we have the following Hardy–Hilbert inequality (cf. [1]):
where the constant factor \(\frac{\pi}{\sin(\pi/p)}\) is the best possible. We still have the following Mulholland inequality with the same best possible constant factor \(\frac{\pi}{\sin(\pi/p)}\) (cf. Theorem 343 of [1], replacing \(\frac{a_{m}}{m}\), \(\frac{b_{n}}{n}\) by \(a_{m}\), \(b_{n}\)):
Inequalities (1)–(2) are important in the analysis and its applications (cf. [1, 2]).
In 2007, Yang [3] first gave a Hilbert-type integral inequality in the whole plane as follows:
where the constant factor \(B(\frac{\lambda}{2},\frac{\lambda}{2})\) (\(\lambda>0\)) is the best possible. Some new results on inequalities (1)–(3) were provided by [4–24]. In 2016, Yang and Chen [25] gave a more accurate extension of (1) in the whole plane as follows:
where the constant factor \(2B ( \lambda_{1},\lambda_{2} ) \) (\(0<\lambda_{1},\lambda_{2}\leq1\), \(\lambda_{1}+\lambda_{2}=\lambda \), \(\xi , \eta\in{[} 0,\frac{1}{2}]\)) is the best possible. Another result was provided by Xin et al. [26].
In this paper, by introducing multi-parameters, applying the weight coefficients and Hermite–Hadamard’s inequality, we give a reverse of the extension of Mulholland’s inequality (2) in the whole plane with the best possible constant factor similar to the reverse of (4). The equivalent forms and a few particular cases are also considered.
2 Some lemmas
In the following, we agree that \(0< p<1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\lambda_{1},\lambda_{2}>0\), \(\lambda_{1}+\lambda_{2}=\lambda\leq 1\), \(\alpha,\beta\in{[} \arccos\frac{1}{3},\frac{\pi}{2}]\), \(\xi ,\eta \in{[} 0,\frac{1}{2}]\), and
Remark 1
In view of the conditions that \(\alpha,\beta\in{[} \arccos\frac{1}{3},\frac{\pi}{2}]\), \(\xi,\eta\in{[} 0,\frac{1}{2}]\), it follows that \((\frac{3}{2}\pm\eta)(1\mp\cos\beta)\geq1\) and \((\frac{3}{2}\pm\xi)(1\mp\cos\alpha)\geq1\).
We set the following functions:
\(B_{\beta}(\eta,y)= \vert y-\eta \vert +(y-\eta)\cos\beta\), and
Definition 1
Define two weight coefficients as follows:
where \(\sum_{ \vert j \vert =2}^{\infty}\cdots =\sum_{j=-2}^{-\infty}\cdots+\sum_{j=2}^{\infty}\cdots\) (\(j=m,n\)).
Lemma 1
(cf. [26])
Suppose that \(g(t)\) (>0) is strictly decreasing in \((1,\infty)\), satisfying \(\int_{1}^{\infty}g(t)\,dt\in \mathbf{R}_{+}\). We have
If \((-1)^{i}g(t)>0\) (\(i=0,1,2\); \(t\in(\frac{3}{2},\infty)\)), \(\int_{\frac {3}{2}}^{\infty}g(t)\,dt\in\mathbf{R}_{+}\), then we have the following Hermite–Hadamard inequality (cf. [27]):
Lemma 2
For \(0<\lambda\leq1\), \(0<\lambda_{2}<1\), the following inequalities are valid:
where
Proof
For \(\vert m \vert \in\mathbf{N}\backslash\{ 1\}\), we set the following functions:
wherefrom
We find
In virtue of \(0<\lambda\leq1\), \(0<\lambda_{2}<1\), we find that for \(y>\frac{3}{2}\),
it follows that \(\frac{H^{(i)}(m,(-1)^{i}y)}{(y-(-1)^{i}\eta)\ln ^{1-\lambda_{2}}[(y-(-1)^{i}\eta)(1+(-1)^{i}\cos\beta)]}\) (\(i=1.2\)) are strictly decreasing and strictly convex in \((\frac{3}{2},\infty)\). By (10) and (13) we find
Setting \(u=\frac{\ln[(y+\eta)(1-\cos\beta)]}{\ln A_{\alpha}(\xi ,m)}\) (\(u=\frac{\ln[(y-\eta)(1+\cos\beta)]}{\ln A_{\alpha}(\xi,m)}\)) in the above first (second) integral, in view of Remark 1, by simplifications, we obtain
By (9) and (13), in the same way, we still have
where \(\theta(\lambda_{2},m)\) is indicated by (12). It follows that \(\theta(\lambda_{2},m)<1\) and
Hence, (11) and (12) are valid. □
In the same way, for \(0<\lambda\leq1\), \(0<\lambda_{1}<1\), we still have the following.
Lemma 3
The following inequalities are valid:
where
Lemma 4
If \((\zeta,\gamma)=(\xi,\alpha)\) (or \((\eta ,\beta)\)), \(\rho>0\), then we have
Proof
Still by (10) we find
By (9), we still can find that
Hence, we prove that (16) is valid. □
3 Main results and a few particular cases
We also define
Theorem 1
Suppose that \(a_{m},b_{n}\geq0\) (\(\vert m \vert , \vert n \vert \in\mathbf{N}\backslash\{1\}\)) and
We have the following reverse equivalent inequalities:
In particular, (i) for \(\alpha=\beta=\frac{\pi}{2}\), \(\xi,\eta\in {}[ 0,\frac{1}{2}]\), setting
we have the following reverse equivalent inequalities:
(ii) For \(\xi,\eta=0\), \(\alpha,\beta\in{[} \arccos\frac{1}{3},\frac{\pi}{2}]\), setting
we have the following equivalent inequalities:
Proof
By the reverse Hölder inequality with weight (cf. [28]) and (8), we find
By (14), in view of \(p-1<0\), it follows that
By (11) and (17), we have (19).
Using the reverse Hölder inequality again, we have
and then by (19) we have (18).
On the other hand, assuming that (18) is valid, we set
and find
By (29) it follows that \(J>0\). If \(J=\infty\), then (19) is trivially valid; if \(J<\infty\), then we have
Hence (19) is valid, which is equivalent to (18).
We have proved that (18) is valid. Then we set
and find
If \(L=0\), then (20) is impossible, namely \(L>0\). If \(L=\infty \), then (20) is trivially valid; if \(L<\infty\), then we have
Hence, (20) is valid.
On the other hand, assuming that (20) is valid, using the reverse Hölder inequality, we have
and then by (20) we have (18), which is equivalent to (20).
Therefore, inequalities (18), (19), and (20) are equivalent.
The theorem is proved. □
Theorem 2
With regards to the assumptions of Theorem 1, the constant factor \(H(\lambda_{1})\) in (18), (19), and (20) is the best possible.
Proof
For \(0<\varepsilon<p\lambda_{1}\), we set \(\widetilde {\lambda }_{1}=\lambda_{1}-\frac{\varepsilon}{p}\) (\(\in(0,1)\)), \(\widetilde {\lambda}_{2}=\lambda_{2}+\frac{\varepsilon}{p}\) (>0), and
If there exists a positive number \(K\geq H(\lambda_{1})\) such that (18) is still valid when replacing \(H(\lambda_{1})\) by K, then, in particular, we have
In view of the above results, it follows that
and then
namely
Hence, \(K=H(\lambda_{1})\) is the best possible constant factor in (18).
The constant factor \(H(\lambda_{1})\) in (19) ((20)) is still the best possible. Otherwise we would reach a contradiction by (30) ((31)) that the constant factor in (18) is not the best possible. □
Remark 2
(i) For \(\xi=\eta=0\) in (22), setting
we have the following new inequality with the best possible constant factor \(\frac{2\pi}{\lambda\sin(\frac{\pi\lambda_{1}}{\lambda})}\):
It follows that (20) ((22)) is an extension of (29).
(ii) If \(a_{-m}=a_{m}\) and \(b_{-n}=b_{n}\) (\(m,n\in\mathbf{N}\backslash\{1\}\)), setting
then (22) reduces to
In particular, for \(\xi=\eta=0\), \(\lambda=1\), \(\lambda_{1}=\lambda _{2}=\frac{1}{2}\) in (33), setting
we have the following reverse Mulholland inequality (2) with the best possible constant π:
4 Conclusions
In this paper, by introducing multi-parameters, applying the weight coefficients, and using Hermite–Hadamard’s inequality, we give a reverse of the extension of Mulholland’s inequality in the whole plane with the best possible constant factor in Theorems 1–2. The equivalent forms and a few particular cases are considered. The technique of real analysis is very important as it is the key to proving the reverse equivalent inequalities with the best possible constant factor. The lemmas and theorems provide an extensive account of this type of inequalities.
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Acknowledgements
This work is supported by the National Natural Science Foundation (Nos. 61370186, 61640222), and the Science and Technology Planning Project Item of Guangzhou City (No. 201707010229). We are grateful for this help.
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BY carried out the mathematical studies, participated in the sequence alignment, and drafted the manuscript. AW participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Wang, A., Yang, B. On a reverse Mulholland’s inequality in the whole plane. J Inequal Appl 2018, 38 (2018). https://doi.org/10.1186/s13660-018-1634-x
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DOI: https://doi.org/10.1186/s13660-018-1634-x