Abstract
By the use of weight coefficients and technique of real analysis, a discrete Hilbert-type inequality in the whole plane with multi-parameters and a best possible constant factor is given. The equivalent forms, the operator expressions, and a few particular inequalities are considered.
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1 Introduction
Suppose that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(a_{m},b_{n}\geq 0\), \(a=\{a_{m}\}_{m=1}^{\infty}\in l^{p}\), \(b=\{b_{n}\}_{n=1}^{\infty}\in l^{q}\), \(\Vert a\Vert _{p}=(\sum_{m=1}^{\infty}a_{m}^{p})^{\frac{1}{p}}>0\), \(\Vert b\Vert _{q}>0\). We have the following well-known Hardy-Hilbert inequality:
where the constant factor \(\frac{\pi}{\sin(\pi/p)}\) is the best possible (cf. [1]). Also we have the following Hilbert-type inequality:
with the best possible constant factor pq (cf. [2]). Inequalities (1) and (2) are important in analysis and its applications (cf. [2–4]).
In 2011, Yang gave an extension of (2) as follows (cf. [5]): If \(0<\lambda_{1},\lambda_{2}\leq1\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(a_{m},b_{n}\geq0\), \(0<\Vert a\Vert _{p,\varphi}=\{\sum_{m=1}^{\infty }m^{p(1-\lambda_{1})-1}a_{m}^{p}\}^{\frac{1}{p}}<\infty \), \(0<\Vert b\Vert _{q,\psi}=\{\sum_{n=1}^{\infty}n^{q(1-\lambda_{2})-1} b_{n}^{q}\}^{\frac{1}{q}}<\infty\), then
where the constant factor \(\frac{\lambda}{\lambda_{1}\lambda_{2}}\) is the best possible.
For \(\lambda=1\), \(\lambda_{1}=\frac{1}{q}\), \(\lambda_{2}=\frac{1}{p}\), inequality (3) reduces to (2). Some other results relate to (1)-(3) are provided by [6–23].
In this paper, by the use of weight coefficients and the technique of real analysis, an extension of (3) in the whole plane is given as follows: For \(0<\lambda_{1},\lambda_{2}\leq1\), \(\lambda_{1}+\lambda _{2}=\lambda\), \(a_{m},b_{n}\geq0\), \(0<\sum_{|m|=1}^{\infty }|m|^{p(1-\lambda _{1})-1}a_{m}^{p}<\infty\), \(0<\sum_{|n|=1}^{\infty}|n|^{q(1-\lambda _{2})-1}b_{n}^{q}<\infty\), we have
Moreover, a generation of (4) with multi-parameters and a best possible constant factor is proved. The equivalent forms, the operator expressions and a few particular inequalities are also considered.
2 Some lemmas
In the following, we agree that \(\mathbf{N}=\{1,2,\ldots\}\), \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\alpha,\beta\in(0,\pi)\), \(\lambda _{1},\lambda _{2}>-\eta\), \(\lambda_{1}+\lambda_{2}=\lambda\), and for \(|x|,|y|>0\),
Lemma 1
(cf. [24])
Suppose that \(g(t)\) (>0) is decreasing in \(\mathbf{R}_{+}\) and strictly decreasing in \([n_{0},\infty)\) (\(n_{0}\in \mathbf{N}\)), satisfying \(\int_{0}^{\infty}g(t)\,dt\in\mathbf{R}_{+}\). We have
Definition 1
Define the following weight coefficients:
where \(\sum_{|j|=1}^{\infty}\cdots=\sum_{j=-1}^{-\infty}\cdots +\sum_{j=1}^{\infty}\cdots\) (\(j=m,n\)).
Lemma 2
If \(\lambda_{2}\leq1-\eta\), then for \(k_{\beta }(\lambda_{1}):=\frac{2(\lambda+2\eta)\csc^{2}\beta}{(\lambda _{1}+\eta )(\lambda_{2}+\eta)}\), we have
where
Proof
For \(|x|>0\), we set
from which we have
We obtain
For fixed \(|m|\in\mathbf{N}\), \(\lambda_{2}\leq1-\eta\), we find that
is decreasing for \(y>0\) and strictly decreasing for \(y\geq\frac {|m|+m\cos \alpha}{1-\cos\beta}\). Under the same assumptions, it is evident that
is decreasing for \(y>0\) and strictly decreasing for \(y\geq\frac {|m|+m\cos \alpha}{1+\cos\beta}\).
Setting \(u=\frac{y(1-\cos\beta)}{|m|+m\cos\alpha}(\frac{y(1+\cos \beta)}{|m|+m\cos\alpha})\) in the above first (second) integral, by simplifications, we find
Still by (11) and (6), we have
We obtain for \(|m|+m\cos\alpha\geq1+\cos\beta\)
In the same way, we have the following.
Lemma 3
If \(\lambda_{1}\leq1-\eta\), then for \(k_{\alpha }(\lambda_{1})=\frac{2(\lambda+2\eta)\csc^{2}\alpha}{(\lambda _{1}+\eta )(\lambda_{2}+\eta)}\), we have
where
Lemma 4
If \(\theta\in(0,\pi)\), then for \(\rho>0\), \(H_{\rho }(\theta):=\sum_{|n|=1}^{\infty}\frac{1}{(|n|+n\cos\theta)^{1+\rho}}\), we have
Proof
We have
By (6), we find
Hence we have (14). □
3 Main results
Theorem 1
If \(\lambda_{1},\lambda_{2}\leq1-\eta\), \(a_{m},b_{n}\geq0\) (\(|m|,|n|\in\mathbf{N}\)),
then we have the following equivalent inequalities:
In particular, for \(\alpha=\beta=\frac{\pi}{2}\), we have the following equivalent inequalities:
Proof
By Hölder’s inequality (cf. [25]) and (8), we have
By (12), we have
By Hölder’s inequality (cf. [25]), we have
On the other hand, assuming that (16) is valid, we set
Then it follows that
By (20), we find \(J<\infty\). If \(J=0\), then (17) is evidently valid; if \(J>0\), then by (16), we have
namely, (17) follows, which is equivalent to (16). □
Theorem 2
As regards the assumptions of Theorem 1, the constant factor \(k_{\alpha,\beta}(\lambda_{1})\) in (16) and (17) is the best possible.
Proof
For any \(\varepsilon\in(0,q(\lambda_{2}+\eta))\), we set \(\widetilde{\lambda}_{1}=\lambda_{1}+\frac{\varepsilon}{q}\) (\(>-\eta\)), \(\widetilde{\lambda}_{2}=\lambda_{2}-\frac{\varepsilon}{q}\) (\(\in(-\eta ,1-\eta)\)), and
If there exists a constant \(k\leq k_{\alpha,\beta}(\lambda_{1})\), such that (16) is valid when replacing \(k_{\alpha,\beta}(\lambda_{1})\) by k, then in particular, we have \(\varepsilon\widetilde {I}<\varepsilon k\widetilde{I}_{1}\), namely,
It follows that
namely,
Hence, \(k=k_{\alpha,\beta}(\lambda_{1})\) is the best possible constant factor of (16).
The constant factor \(k_{\alpha,\beta}(\lambda_{1})\) in (17) is still the best possible. Otherwise, we would reach a contradiction by (21) that the constant factor in (16) is not the best possible. □
4 Operator expressions
We set functions \(\Phi(m)\) and \(\Psi(n)\) as follows:
from which we have
We also set the following weight normed spaces:
Then for \(a=\{a_{m}\}_{|m|=1}^{\infty}\in l_{p,\Phi }\), \(c=\{c_{n}\}_{|n|=1}^{\infty}\), \(c_{n}=\sum_{|m|=1}^{\infty }k(m,n)a_{m}\), in view of (17), we have \(\Vert c\Vert _{p,\Psi^{1-p}}< k_{\alpha,\beta }(\lambda_{1})\Vert a\Vert _{p,\Phi}<\infty\), namely, \(c\in l_{p,\Psi^{1-p}}\).
Definition 2
Define a Hilbert-type operator \(T:l_{p,\Phi }\rightarrow l_{p,\Psi^{1-p}}\) as follows: For any \(a=\{a_{m}\} _{|m|=1}^{\infty}\in l_{p,\Phi}\), there exists a unique representation \(c=Ta\in l_{p,\Psi^{1-p}}\). We also define the formal inner product of Ta and \(b=\{b_{n}\}_{|n|=1}^{\infty}\in l_{q,\Psi}\) (\(b_{n}\geq0\)) as follows:
Then for \(a_{m}\geq0\) (\(|m|\in\mathbf{N}\)), we may rewrite (16) and (17) as follows:
We define the norm of operator T as follows:
Then \(\Vert Ta\Vert _{p,\Psi^{1-p}}\leq\Vert T\Vert \cdot\Vert a\Vert _{p,\Phi}\). Since by Theorem 2, the constant factor \(k_{\alpha,\beta}(\lambda_{1})\) in (24) is the best possible, we have
Remark 1
(i) For \(\eta=0\), (16) reduces to the following inequality:
In particular, for \(\alpha=\beta=\frac{\pi}{2}\), (27) reduces to (4). If \(a_{-m}=a_{m}\), \(b_{-n}=b_{n}\) (\(m,n\in\mathbf{N}\)), then (4) reduces to (3). Hence, (16) is an extension of (4) with multi-parameters.
(ii) For \(\eta=-\lambda\), \(-1\leq\lambda_{1}\), \(\lambda_{2}<0\) in (16), we have
In particular, for \(\alpha=\beta=\frac{\pi}{2}\), we have
(iii) For \(\lambda=0\) in (16), we have \(\lambda_{2}=-\lambda _{1}\), \(|\lambda_{1}|<\eta\) (\(\eta>0\)) and
In particular, for \(\alpha=\beta=\frac{\pi}{2}\), we have
The above particular inequalities are all with the best possible constant factors.
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Acknowledgements
This work is supported by the Science and Technology Planning Project of Guangdong Province (No. 2013A011403002), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25).
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BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. DX and QC participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Xin, D., Yang, B. & Chen, Q. A discrete Hilbert-type inequality in the whole plane. J Inequal Appl 2016, 133 (2016). https://doi.org/10.1186/s13660-016-1075-3
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DOI: https://doi.org/10.1186/s13660-016-1075-3